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7 Awesome Pattern Backgrounds for Your Slides and How to Create Them in PowerPoint

Anuj Malhotra

What differentiates a professional design from an amateur design? The BIG idea or the creative idea, yes. But often, it is the focus on the SMALL details that truly give a design a professional touch.

One such small detail that you can also incorporate in your presentation slides and other designs is a Pattern Background . What’s the big deal about a design background? It is relegated to the background, buried behind the content and is too light to draw any special attention to itself. So why waste so much time in designing it in the first place?

Because in design, everything counts. The full stop or the absence of it, the three dots giving continuity or the exclamation mark screaming for attention, the small icon in the corner of the slide or the shadow you gave to an object...everything counts. Backgrounds, especially pattern backgrounds, add meaning to the slide. It’s function is not just to make the slide look pretty (though it achieves in doing that too) but also to give the slide a visual impetus by giving a context to the message.

If the pattern you choose is relevant to the presentation topic, it adds POWER to your presentation. How? By giving credibility to the message that you put in the foreground. The message sitting on a plain white slide is all right, it is common and there is nothing so special about it. But the same message on a painted canvas has a different personality altogether.

Let’s give our slides the personality it deserves by using creative pattern backgrounds. We’ll show you 7 amazing pattern backgrounds (includes geometric pattern backgrounds and abstract pattern backgrounds ) that you can try in your upcoming presentations and also teach you how to create them in PowerPoint without shelling a single penny.

Let’s get started:

Free Pattern Backgrounds for Slides & How to Create Them

#1- diamond pattern background.

“A diamond is forever.” This rare and beautiful stone symbolizes everlasting. Using this exquisite icon, therefore, gives a classic and “superior” look to the design. The diamond shape provided in the Shapes menu in PowerPoint shows four sides joining together- said to symbolize creativity that comes by joining the four elements of nature- earth, fire, air, and water. Give a rich look to your presentation design with this subtle yet powerful pattern background:

Steps to Create This Presentation Background:

• Go to the Insert tab> Shapes menu> locate the Diamond shape (see the screenshot below)
• Simply click on the slide to have the diamond shape drawn
• Reduce it’s size to .69 from the Format tab
• Right click on the shape and select Format Shape
• The Format Shape window will open to your right. Choose No Fill and a Solid line of the color of your choice. Under Dash type , select the Square dot option (see the below screenshot)

We now have to create a row of diamonds from the left corner to right corner of the slide. Rather than copy and paste the diamond shape, easier way is to press Ctrl and Shift together and drag the diamond and place it next to the previous. When you have placed 3-4 diamonds, group them by using the shortcut- Ctrl + G and drag and place them next to the previous group. This will help you create the row faster. When you are done, group the row by using the Ctrl + G shortcut again.

Now, press Ctrl and Shift and drag this row. Place one row below touching the above row and do it till your complete slide is covered with diamond rows (see screenshot below for reference):

Group all the rows together and your pattern background is ready! You want your text to be readable, so give it a transparency level as per the requirements. The transparency level can be s high as 80-90%. To give this transparency, right click on the pattern > Format Shape > Line > Transparency .

Another technique is to create pattern on the blank white slide. Add a colored rectangular shape over the slide (grey or blue or whatever presentation colors you use) and give the rectangle transparency. The effect will be the same. You can now use this background to create artistic slides like the one below:

Without the pattern, the slide would still have looked great. But pattern adds a subtle but powerful effect to the slide. In fact, it enhances the meaning of the audience. The majestic pyramid and the majestic diamonds complement each other. The audience won’t know why they love the slide so much, but you know. Each element, as small as a slightly visible pattern or the line below the heading, adds value to the overall design.

#2- Hexagon Pattern Background

Hexagon is another shape that attracts a lot of interest in symbolism. It indicates balance and unity. This geometric shape with six edges and six vertices has inspired in architecture design, tattoos, art and fashion. Using this presentation background will lend credibility to your design and the content. Here’s how an interwoven hexagon pattern looks:

Steps to Create this Pattern Background:

• Go to the Insert tab> Shapes menu> locate the Hexagon shape (see the screenshot below)
• Simply click on the slide to have the hexagon shape drawn
• Reduce it’s size to .42- Shape Height and .49- Shape Width from the Format tab
• Open the Rotate menu and click Rotate Right 90 to have the pointed edge of the hexagon facing upward (see the screenshot below)
• The Format Shape window will open to your right. Choose No Fill and a Solid line of the color of your choice

Press Ctrl and Shift to create a row of hexagons from corner left to corner right (step 4 in the screenshot below). Use the Ctrl + G shortcut to make the entire row one object. Unlike the diamond pattern, in hexagon we won’t put each row directly below the other. Copy the top row and place the duplicate in such a way that each hexagon covers the gaps of the above row (see step 5 in the screenshot below for quick understanding):

Do this for all the rows below and your pattern background will finally look like this:

Group all the rows and give the pattern transparency so that your message stands out from the background. Here’s a business slide that applies this background to create an amazing effect. The consistency in the pattern and the shape used for the image helps create the visually stunning effect:

#3- Networking Pattern Background

Any Information Technology presentation or a supply chain presentation showing networking among different systems can apply the networking pattern to turn a dull presentation into a visually impactful one. Whether you are talking about global communications or social media networks, this pattern can help you establish the context and add power to your content.

• Insert the Oval shape from the Shapes menu ( Insert tab)
• Press Ctrl and Shift to draw a perfectly symmetrical circle on the slide or simply click on the slide and a perfectly symmetrical circle will appear on the slide
• Right click on the circle > Format Shape > Solid Fill- White > Line- No line
• Create 3 circles of varying sizes. Go to the Format tab and in the Size group, specify the following dimensions of each circle- Circle 1, Height and Width- 0.75 , Circle 2- 0.50 and Circle 3- 0.30 (see the screenshot below for reference)

Distribute the circles across the slide in a haphazard manner like we did below (circles highlighted yellow and blue just for reference). The Circles 1 have been colored yellow just for your ease of understanding. Keep around 4 or 5 of circles 1 around the slide. The Circles 2 have been highlighted blue just for reference. Distribute around 10-12 of these circles around the slide. The color of circles 3 has not been changed and their number is almost the double of circles 2. So, your slide will look like a messy arrangement of bubble shapes like the one below:

Next, connect all the circles with a line. Go to the Insert tab > Shapes menu and locate the Line . From the Format Shape menu, you can customize the color of the line. We have kept it white. Start connecting 2 circles with a line.  When you connect circles, you’ll find PowerPoint highlights the boundaries of the circles. This is to help you connect the circles easily and more effectively (see the screenshot below):

To make the pattern look even more striking, we’ll add dotted circles around the bigger circles (circles 1 and 2 highlighted yellow and blue respectively just for reference). Creating dotted circles is very easy- Insert the oval shape > right click on the shape > Format Shape > No fill > Solid line > Width- 1 point > Dash type- Dotted round . This is how the pattern now looks:

Obviously, we have to give the pattern a transparency so that the content we put in front is easily readable. On way is to give the circles and lines a transparency level of 80-90% from the Format Shape window. Second and easier way is to insert a rectangle over the complete slide and give it a solid fill color of your choice. Give the rectangle transparency which will automatically make the pattern fade in the background.

To create a breathtaking Title slide or section header slide, you can add a gradient background over the pattern slide and make the slide look as if designed by a professional designer:

Want to download this amazing gradient background for your presentations? Download this and 7 amazing gradient backgrounds for free from the blog post below:

Download Gradient Backgrounds for Free

Here’s the pattern background lending visual support to a Supply Chain Management slide:

#4- Circular Pattern Background

Circle represents infinity, wholeness, totality and eternity. But it can also be perceived as boring. So we won’t be taking a regular circle. We want to create dynamic slides. Let’s take an arc instead and extend it so that it appears to be a broken circle. Here’s such a pattern background (looks celestial, doesn’t it?)

• Insert the Block Arc shape from the Shapes menu (under the Insert tab)
• On selecting the shape, 2 yellow handles will appear
• Drag the right yellow handle towards the right direction which will make the arc thickness lesser
• Drag the left handle towards bottom and towards the first yellow handle which will make the arc’s length bigger (see the screenshot below):

Right click on the arc > Format Shape. Remove the outline from the arc and make the solid fill color white. Duplicate the shape using Ctrl C and Ctrl V shortcut. Reduce the size of the duplicate arc and place it in the centre of the bigger arc (see the screenshot below). Rotate the smaller arc using the rotation handle on top such that the gaps of the arcs are facing in the opposite direction:

Group the bigger and smaller arc by using the shortcut- Ctrl + G . Copy this grouped object and reduce its size using the sizing handles. Distribute all these arcs around the slide to have your pattern design ready in 2 minutes! Don’t forget to give the pattern a transparency of 70-80% for easy readability of the content you put on the slide.

It’s not necessary to use pattern across the complete slide canvas. You can put it in one half or area of the slide. Here’s a channel marketing slide that uses this pattern:

#5- Growth Trend Pattern Background

Quarterly Business Review, Quarterly Sales Review presentation, Annual Report, market analysis deck or any presentation that talks about trends can benefit from growth pattern background. This pattern consist of a magnifying glass that shows the focus on market and a line graph showcasing a trend. Not is it just very easy to create but also lends credibility to your presentation.

• Insert the Circle: Hollow shape from the Shapes menu (under the Insert tab)
• Select the shape and drag the yellow handle towards left to increase the hollow space and reduce the width of the circle (see the screenshot below)
• Right click on the shape > Format Shape > Solid Fill > Color- White > Line- No line

We now need to create a handle so that the shape looks like a magnifying glass. To do so, add a rectangle from the Shapes menu and place it below the hollow circle. Tilt it a bit using the rotation handle on the top. Give the same formatting to the rectangle as given to the circle- white fill and no outline. Lastly, add a $ text symbol in the centre of the circle. Press Shift and 4 to add the $ sign right from your keyboard.

Learn to Create More Magnifying Glass Designs

We now need to show an upward moving line graph. To create, simply insert lines from the Shapes menu. Give it a 4 point width and white color from the Format Shape window. Place lines downward and upward alternatively to show a rising and falling graph. The last line should be facing upward if you want to show a growth trend through your pattern. Also add circles at the intersection of two lines to make the line graph more visually attractive. Your pattern background is now ready!

Add a transparent layer over the pattern to easily display your presentation content. You can also minimise the pattern as done in the slide below:

#6- Gears Pattern Background

Any business presentation outlining operations and procedures can look impressive and professional by using the gears pattern background. Since gears represent teamwork, interlinked processes, unity in diversity, smooth operations and so on, they are a perfect fit for many presentations and industries. Either you can show multiple gears interlinked together…

Or a single large gear to keep the pattern minimal like the one below:

Steps to Create a Gear in PowerPoint:

We have a complete post on how to create Gears in PowerPoint . Here’s a quick summary of the steps to create a gear in just 2 minutes:

• Insert the Oval shape from the Shapes menu (Press Shift to draw a perfectly symmetrical circle)
• Right click on the circle and click Format Shape
• Under Fill , select the No fill radio button
• Under Line , select Solid line > Color- White
• Increase the Width of the line to 35 points
• Change the Dash type to Round Dot (see the screenshot below)

• Add the Hollow: Circle shape from the Shapes menu
• Place the donut in the centre of the dotted circle
• Move the yellow dot to left to adjust the thickness (see the screenshot below)

Group the dotted circle and the donut using the shortcut Ctrl + G and the gear is ready! Use it to turn boring operational slides into exciting ones like this:

#7- Lines Pattern Background

Abstract pattern backgrounds can also light up your slides and give a creative look to your presentation. One such abstract pattern that you can easily create yourself is with lines. Criss-crossing lines signify a network and a complex web of relationships and processes.

• Insert Line from the Shapes menu (Press Shift to draw a straight line)
• Right click> Format Shape
• Go to the Line menu > End Arrow type > choose the Oval Arrow option

Add a gradient effect towards the left corner of the slide. To do that, insert any shape from the Shapes menu (we chose the Oval shape). Right click on the shape and click Format Background . Select the Gradient fill option> Type- Radial > Direction- From centre (see the screenshot below). Colors of the background should match the slide background. Place this shape towards the corner of the slide.

The abstract line pattern is ready that you can use in any presentation. Here’s an example:

Which pattern background did you like the most? Leave your feedback in the comments below for we’ll love to know if you found this presentation tutorial useful.

Related posts:

• How to Quickly Find the Best Content for Your Presentation on SlideTeam
• How To Create an Awesome PowerPoint Presentation in 3 Steps
• How Icons Can Give a Complete Makeover to Your Slides & Make Them Look Sexy!
• Why Bullet Points Are Not That Bad After All!

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Free Pattern PowerPoint Templates

Free Abstract Neon Light PowerPoint Template

Free Abstract Neon Light PowerPoint Template is a striking presentation template that is compatible with both PowerPoint and Google Slides. This PPT template features an abstract background design, characterized by neon lights that create a mesmerizing network effect, making it a perfect choice for technology-related presentations and other creative use cases. The PowerPoint background in …

Add an artistic touch to your idea by choosing one of the patterns available on FPPT, such as Color Strip and Blue Widescreen. Choose from a number of free PowerPoint slides and pick one that caters to your theme.

Download Patterns PPT backgrounds with a Black & White and Imperial theme on Microsoft PowerPoint. Alternatively, you can download other free pattern backgrounds for PowerPoint with editable styles.

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326 Best Pattern-Themed Templates for PowerPoint & Google Slides

With over 6 million presentation templates available for you to choose from, crystalgraphics is the award-winning provider of the world’s largest collection of templates for powerpoint and google slides. so, take your time and look around. you’ll like what you see whether you want 1 great template or an ongoing subscription, we've got affordable purchasing options and 24/7 download access to fit your needs. thanks to our unbeatable combination of quality, selection and unique customization options, crystalgraphics is the company you can count on for your presentation enhancement needs. just ask any of our thousands of satisfied customers from virtually every leading company around the world. they love our products. we think you will, too" id="category_description">crystalgraphics creates templates designed to make even average presentations look incredible. below you’ll see thumbnail sized previews of the title slides of a few of our 326 best pattern templates for powerpoint and google slides. the text you’ll see in in those slides is just example text. the pattern-related image or video you’ll see in the background of each title slide is designed to help you set the stage for your pattern-related topics and it is included with that template. in addition to the title slides, each of our templates comes with 17 additional slide layouts that you can use to create an unlimited number of presentation slides with your own added text and images. and every template is available in both widescreen and standard formats. with over 6 million presentation templates available for you to choose from, crystalgraphics is the award-winning provider of the world’s largest collection of templates for powerpoint and google slides. so, take your time and look around. you’ll like what you see whether you want 1 great template or an ongoing subscription, we've got affordable purchasing options and 24/7 download access to fit your needs. thanks to our unbeatable combination of quality, selection and unique customization options, crystalgraphics is the company you can count on for your presentation enhancement needs. just ask any of our thousands of satisfied customers from virtually every leading company around the world. they love our products. we think you will, too.

Widescreen (16:9) Presentation Templates. Change size...

Blue wave pattern on black background

Transparent glossy globe on top of golden wave pattern orange and yellow background

Black silhouettes of dancing people on abstract floral pattern with disco lights on red and black background

Abstract purple, orange, gold wave pattern on dark background

Silhouette of people standing on globe pattern hold hands

Five opened lilac pink callas over delicate floral pattern on green and black background

Transparent Planet Earth on a blue circuit pattern background with black lines and white dots

Two terrain roads separated in different directions between forest trees, hiking trails in forest with floral white pattern top and bottom

Earth globe with jigsaw puzzle pattern and cuttings from famous books

Abstract flower in light, stars on galaxy sky, blue, pink, black background

Blue and white abstract wave pattern

3D dark colored tiles with old letterpress white letters depicting the message the big idea and exclamation mark on red painted pattern background

Christmas theme with golden bells and red Christmas globes and golden mistletoes, with golden abstract pattern on red background

Abstract blue wave pattern with bright white snowflakes

Floral pattern with six various sized purple flowers on gradient light purple and white background

Plain blue, tan, and white line pattern

Gray metal spheres placed on blue-white checkers pattern surface

Eagle soaring into blue sky with American flag pattern on body

Abstract circle and line pattern on blue and green, suggesting sunrise over green grass

A beautiful pattern with its mild version in the background

Theme enhanced with vintage concept conceptual old retro aged paper texture isolated on white background abstract damaged parchment or label as a banner for grunge ornament book letter time pattern history designs

Abstract floral pattern with various colored shapes and glowing colored spots on a black background

Slides consisting of winter landscape branches form a heart-shaped pattern backdrop

Abstract depiction of checked box pattern on a blue background

PPT theme consisting of traditional moroccan ornament

Glowing earth globe over floral pattern in blue background

Slide deck with fine brushed metal with reflection photograph of brushed metal or hair line pattern metal high resolution sharp to the corners

Geometric multicolored peacock tail pattern on dark background

A representation of a pattern with blackish background

Five puppies little dogs sitting on a square pattern mattress black couch

White floral pattern frame on gradient red orange and yellow background with sunrays

A moving pattern in the background - widescreen format

A floral pattern in the background with greenish outline

Close up of rock pattern background, depiction of rock

Electric green sine waves on background of square pattern

Abstract floral pattern design with red flowers on white background

Black colored world map outline on blue printed circuit board pattern

Presentation design enhanced with top view of rows blackcurrant bushes cultivated land aerial photography drone shot agricultural area of ukraine agrarian industry artistic wallpaper abstract natural pattern beauty of earth

PPT layouts having vintage pattern on an arab carpet background

Theme featuring scottish tartan pattern part of a traditional kilt background

More pattern templates for powerpoint and google slides:.

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20 Great Examples of PowerPoint Presentation Design [+ Templates]

Published: January 17, 2024

When it comes to PowerPoint presentation design, there's no shortage of avenues you can take.

While all that choice — colors, formats, visuals, fonts — can feel liberating, it‘s important that you’re careful in your selection as not all design combinations add up to success.

In this blog post, I’m sharing some of my favorite PowerPoint tips and templates to help you nail your next presentation.

Table of Contents

What makes a good PowerPoint presentation?

Powerpoint design ideas, best powerpoint presentation slides, good examples of powerpoint presentation design.

In my opinion, a great PowerPoint presentation gets the point across succinctly while using a design that doesn't detract from it.

Here are some of the elements I like to keep in mind when I’m building my own.

1. Minimal Animations and Transitions

Believe it or not, animations and transitions can take away from your PowerPoint presentation. Why? Well, they distract from the content you worked so hard on.

A good PowerPoint presentation keeps the focus on your argument by keeping animations and transitions to a minimum. I suggest using them tastefully and sparingly to emphasize a point or bring attention to a certain part of an image.

2. Cohesive Color Palette

I like to refresh my memory on color theory when creating a new PowerPoint presentation.

A cohesive color palette uses complementary and analogous colors to draw the audience’s attention and help emphasize certain aspects at the right time.

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It‘s impossible for me to tell you the specific design ideas you should go after in your next PowerPoint, because, well, I don’t know what the goal of your presentation is.

Luckily, new versions of PowerPoint actually suggest ideas for you based on the content you're presenting. This can help you keep up with the latest trends in presentation design .

PowerPoint is filled with interesting boilerplate designs you can start with. To find these suggestions, open PowerPoint and click the “Design” tab in your top navigation bar. Then, on the far right side, you'll see the following choices:

This simplistic presentation example employs several different colors and font weights, but instead of coming off as disconnected, the varied colors work with one another to create contrast and call out specific concepts.

What I like: The big, bold numbers help set the reader's expectations, as they clearly signify how far along the viewer is in the list of tips.

10. “Pixar's 22 Rules to Phenomenal Storytelling,” Gavin McMahon

This presentation by Gavin McMahon features color in all the right places. While each of the background images boasts a bright, spotlight-like design, all the characters are intentionally blacked out.

What I like: This helps keep the focus on the tips, while still incorporating visuals. Not to mention, it's still easy for me to identify each character without the details. (I found you on slide eight, Nemo.)

11. “Facebook Engagement and Activity Report,” We Are Social

Here's another great example of data visualization in the wild.

What I like: Rather than displaying numbers and statistics straight up, this presentation calls upon interesting, colorful graphs, and charts to present the information in a way that just makes sense.

12. “The GaryVee Content Model,” Gary Vaynerchuk

This wouldn‘t be a true Gary Vaynerchuk presentation if it wasn’t a little loud, am I right?

What I like: Aside from the fact that I love the eye-catching, bright yellow background, Vaynerchuk does a great job of incorporating screenshots on each slide to create a visual tutorial that coincides with the tips. He also does a great job including a visual table of contents that shows your progress as you go .

13. “20 Tweetable Quotes to Inspire Marketing & Design Creative Genius,” IMPACT Branding & Design

We‘ve all seen our fair share of quote-chronicling presentations but that isn’t to say they were all done well. Often the background images are poor quality, the text is too small, or there isn't enough contrast.

Well, this professional presentation from IMPACT Branding & Design suffers from none of said challenges.

What I like: The colorful filters over each background image create just enough contrast for the quotes to stand out.

14. “The Great State of Design,” Stacy Kvernmo

This presentation offers up a lot of information in a way that doesn't feel overwhelming.

What I like: The contrasting colors create visual interest and “pop,” and the comic images (slides 6 through 12) are used to make the information seem less buttoned-up and overwhelming.

15. “Clickbait: A Guide To Writing Un-Ignorable Headlines,” Ethos3

Not going to lie, it was the title that convinced me to click through to this presentation but the awesome design kept me there once I arrived.

What I like: This simple design adheres to a consistent color pattern and leverages bullet points and varied fonts to break up the text nicely.

16. “Digital Transformation in 50 Soundbites,” Julie Dodd

This design highlights a great alternative to the “text-over-image” display we've grown used to seeing.

What I like: By leveraging a split-screen approach to each presentation slide, Julie Dodd was able to serve up a clean, legible quote without sacrificing the power of a strong visual.

17. “Fix Your Really Bad PowerPoint,” Slide Comet

When you‘re creating a PowerPoint about how everyone’s PowerPoints stink, yours had better be terrific. The one above, based on the ebook by Seth Godin, keeps it simple without boring its audience.

What I like: Its clever combinations of fonts, together with consistent color across each slide, ensure you're neither overwhelmed nor unengaged.

18. “How Google Works,” Eric Schmidt

Simple, clever doodles tell the story of Google in a fun and creative way. This presentation reads almost like a storybook, making it easy to move from one slide to the next.

What I like: This uncluttered approach provides viewers with an easy-to-understand explanation of a complicated topic.

19. “What Really Differentiates the Best Content Marketers From The Rest,” Ross Simmonds

Let‘s be honest: These graphics are hard not to love. I especially appreciate the author’s cartoonified self-portrait that closes out the presentation. Well played, Ross Simmonds.

What I like: Rather than employing the same old stock photos, this unique design serves as a refreshing way to present information that's both valuable and fun.

20. “Be A Great Product Leader,” Adam Nash

This presentation by Adam Nash immediately draws attention by putting the company's logo first — a great move if your company is well known.

What I like: He uses popular images, such as ones of Megatron and Pinocchio, to drive his points home. In the same way, you can take advantage of popular images and media to keep your audience engaged.

PowerPoint Presentation Examples for the Best Slide Presentation

Mastering a PowerPoint presentation begins with the design itself.

Get inspired by my ideas above to create a presentation that engages your audience, builds upon your point, and helps you generate leads for your brand.

Editor's note: This post was originally published in March 2013 and has been updated for comprehensiveness. This article was written by a human, but our team uses AI in our editorial process. Check out our full disclosure to learn more about how we use AI.

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Patterns and sequences

Aug 14, 2014

2.04k likes | 4.3k Views

Patterns and sequences. We often need to spot a pattern in order to predict what will happen next. In maths, the correct name for a pattern of numbers is called a SEQUENCE.

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• corresponding arithmetic series
• first term t1
• first night
• quadratic sequences
• arithmetic series

Presentation Transcript

Patterns and sequences We often need to spot a pattern in order to predict what will happen next. In maths, the correct name for a pattern of numbers is called a SEQUENCE. The first number in a SEQUENCE is called the FIRST TERM T1; the second is the SECOND TERM T2, the nth term is Tn,and so on.

Patterns and sequences For any pattern it is important to try to spot what is happening before you can predict the next number. The first 2 or 3 numbers is rarely enough to show the full pattern - 4 or 5 numbers are best.

Patterns and sequences For any pattern it is important to try to spot what is happening before you can predict the next number. 1, 2, …… What’s the next number?

Patterns and sequences For any pattern it is important to try to spot what is happening before you can predict the next number. 1, 2, 4,… Who thought that the next number was 3? What comes next?

Patterns and sequences For any pattern it is important to try to spot what is happening before you can predict the next number. 1, 2, 4, 8, 16, … What comes next?

Patterns and sequences Look at what is happening from 1 TERM to the next. See if that is what is happening for every TERM. 5, 8, 12, 17, 23, …, … + 3

Patterns and sequences Look at what is happening from 1 TERM to the next. See if that is what is happening for every TERM. 5, 8, 12, 17, 23, …, … + 3 + 3 X

Patterns and sequences Look at what is happening from 1 TERM to the next. See if that is what is happening for every TERM. 5, 8, 12, 17, 23, …, … + 3 + 4

Patterns and sequences Look at what is happening from 1 TERM to the next. See if that is what is happening for every TERM. 5, 8, 12, 17, 23, …, … + 3 + 4 + 5 

Patterns and sequences Look at what is happening from 1 TERM to the next. See if that is what is happening for every TERM. 5, 8, 12, 17, 23, …, … + 3 + 4 + 5 + 6 

Patterns and sequences Look at what is happening from 1 TERM to the next. See if that is what is happening for every TERM. 5, 8, 12, 17, 23, 30, … + 3 + 4 + 5 + 6 + 7

Patterns and sequences Now try these patterns: 3, 7, 11, 15, 19, …, … 128, 64, 32, 16, 8, …, … 1000, 100, 10, 1, …, … 5, 15, 45, 135, …, …

Arithmetic Sequence • A sequence, in which any term after the first can be obtained by adding a fixed number to the term before it, is called an arithmetic sequence. Term 1 ( T1) = a The fixed number is called the common difference and is represented by the letter d. T1 (a) = 2 d = 2

The nth term of an arithmetic sequence When using a number sequence we sometimes need to know, for example, the 50th or 1000th term. To do this we need to find a rule that generates the sequence • Example : If the nth term of a sequence is 4n – 3, write down the first 5 terms of the sequence. • Tn = 4n -3 • T1 = 4(1) -3 = 1 • T2 = 4(2) -3 = 5 • T3 = 4(3) – 3 =9 • T4 = 4(4) – 3 = 13 • T5 = 4(5) – 3 = 17

Finding the nth term of an arithmetic sequenceTn = a + ( n -1)d In an arithmetic sequence 3, 8, 13,……. Find (i) a (ii) d (iii) Tn (iv) T20 • a = T1 = 3 • d = common difference = 5 • Tn = a + (n -1)d = 3 + (n – 1)5 = 3 + 5n – 5 = 5n – 2 (iv) T20 = 5(20) – 2 = 100 – 2 = 98

(i) Find the nth term of the arithmetic sequence 7, 10, 13, 16 Tn = a + ( n – 1) d 7 + ( n – 1 ) 3 7 + 3n – 3 3n + 4 (ii) Which term of the sequence is 97 ? 3n + 4 = 97 3n = 93 n = 31 (iii) Show that 168 is not a term of the sequence. 3n + 4 = 168 3n = 164 n = 164 / 3, since n is not a whole number, 168 is not a term in the sequence.

Finding the values of a and d If we are given any two terms of an arithmetic sequence, we can use simultaneous equations to find the value of a and d. Example: T4 of an arithmetic sequence is 11 and T9 = 21 Find the values of a and d and hence find T50. Tn = a + (n – 1)d T4 : a + 3d = 11 T9 : a + 8d = 21 By using simultaneous equations, a = 5 and d = 2

Arithmetic Series • The African-American tribal celebration involves the lighting of candles every night for seven nights. The first night one candle is lit and blown out.

Arithmetic Series • The second night a new candle and the candle from the first night are lit and blown out. The third night a new candle and the two candles from the second night are lit and blown out.

Arithmetic Series • This process continues for the seven nights. • We want to know the total number of lightings during the seven nights of celebration.

Arithmetic Series • The first night one candle was lit, the 2nd night two candles were lit, the 3rd night 3 candles were lit, etc. • So to find the total number of lightings we would add: • 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28

Arithmetic Series • Arithmetic sequence: 2, 4, 6, 8, 10 • Corresponding arithmetic. series:2 + 4 + 6 + 8+10 • Arithetic Sequence: -8, -3, 2, 7 • Corresponding Arithmetic Series: -8 + -3 + 2 + 7

Arithmetic Series • Sn is the symbol used to represent the first ‘n’ terms of a series. • Given the sequence 1, 11, 21, 31, 41, 51, 61, 71, … find S4 • We simply add the first four terms 1 + 11 + 21 + 31 = 64

Arithmetic Series • Find S8 of the arithmetic sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 =36

Arithmetic Series What if we wanted to find S100 for the sequence in the last example. It would be very long to have to list all the terms and try to add them up. • Let’s figure out a formula!! :)

Arithmetic Series When the terms of an arithmetic sequence are added, they form an arithmetic series. Example 1,3,5,7 …….is an arithmetic sequence 1+3+5+7 ….is an arithmetic series. We use Sn to show the sum of the first n terms. We use the following formula to find the sum of n terms of an arithmetic series.

Quadratic Sequences • A sequence is a set of numbers that are connected in some way. In this section we will look at quadratic sequences where the difference between the terms changes. • Consider the following sequence : • Here, the differences between terms are not constant, but there is still a pattern. • the differences between each number in the sequence vary • But the second difference is a constant 2. • When the second difference is constant, you have a quadratic sequence - ie, there is an n2 term.

Quadratic Sequences Learn these rules: If the second difference is 2, you start with n2. If the second difference is 4, you start with 2n2. If the second difference is 6, you start with 3n2.

Finding the nth term of a quadratic sequence Consider the sequence 3, 10, 21, 36 Work out each difference 7 11 15 diff is not constant Find the difference of the difference 4 4 As the difference of the difference is constant, it is a quadratic sequence Tn = an² + bn + c……. Now write T1 and T2 in terms of a and b T1 = 2(1)² + b(1) + c…..a = 2 because it is half the second difference 2 + b + c = 3 ………because we know from the sequence that T1 = 3 b + c = 3 – 2 b + c = 1 ( This is equation 1) Now do the same for T2 Tn = an² + bn + c T2 = 2(2) ² + b(2) + c = 8 + 2b + c = 10……..because we know from the sequence T2 = 10 = 2b + c = 2 ……….. ( This is equation 2 ) You now solve for b and c, by means of Simultaneous Equations. B = 1 anc c = 0

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