Double Trouble

8096 /www/nrich/html/content/id/8096/ 1 0000-00-00T00:00:00 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><h3>Charlie has been adding fractions in the sequence $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots$ where each fraction is half the previous one:</h3> $$\frac{1}{2} + \frac{1}{4} $$ $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8}$$ $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} +\frac{1}{16}$$ Work out the answers to Charlie's sums. What do you notice? Will the pattern continue? How do you know? Try writing an expression for $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{2^n}$$ Could you convince someone else that your expression is correct for all values of $n$? Charlie drew a diagram to try to explain what was going on: <iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/ojexFq3xU0E" width="420"></iframe> Use Charlie's diagram to explain why $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{2^n} = 1-\frac{1}{2^n} = \frac{2^n-1}{2^n}$$ <h3>Alison has been adding numbers in the sequence $1, 2, 4, 8, \dots$ where each number is twice the previous one:</h3> $$1 + 2$$ $$1 + 2 + 4$$ $$1 + 2 + 4 + 8$$ Work out the answers to Alison's sums. What do you notice? Will the pattern continue? How do you know? Try writing an expression for $$1 + 2 + 4 + \dots + 2^n$$ Could you convince someone else that your expression is correct for all values of $n$? Alison drew a diagram to try to explain what was going on: <iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/kHpSoLgFWHw" width="420"></iframe> Can you use Alison's diagram to explain why $$1 + 2 + 4 + \dots + 2^n = 2^{n+1}-1$$</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">This is a nice problem - it's great that you can really see what's going on! Let's see how you fared with it... Hani, from the British School, started off by spotting a pattern: 1/2 + 1/4 = 3/4 1/2 + 1/4 + 1/8 = 7/8 1/2 + 1/4 + 1/8 + 1/16 = 15/16 In these sums, the denominator of the answer is the denominator of the last fraction in the sum, and the numerator is one less than the denominator. Great! The class at Toongabbie Public School spotted this too. Well done! Ayesha, from Westfield Middle School, instead noticed something geometric: In Charlie's diagram, if you draw the sum in a square, then for each new term of the sum you add, half of the remaining gap is halved again. Zachary, from Sonoran Science Academy, wrote the following: First, let us see what we are faced with. We want to simplify the expression: $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{2^n}$. Let's make the denominators the same. We will now see that the expression is: $\frac{2^{n-1}}{2^n} + \frac{2^{n-2}}{2^n} + \dots + \frac{1}{2^n}$. Now, we can rearrange the terms to get: $\frac{1+2+4+8+\dots +2^{n-1}}{2^n}$. This may seem like a hard term to simplify, but let us just look at the numerator of the term. For small values of n: n=1 -&gt; 1 n=2 -&gt; 1+2= 3 n=3 -&gt; 1+2+4= 7 n=4 -&gt; 1+2+4+8= 15 It's easy to spot the pattern here: each value of n gives us $2^n - 1$. Now we have a nice simplified term: $\frac{2^n - 1}{2^n}$. Nice, but how do you know that the pattern you've spotted will continue? What about Alison's sum? Rajeev, from Haberdashers' Aske's Boys' School, noticed: I notice that the answer was 1 less than the next number in the sequence, e.g. 1+2+4+8 = 15, which is one less than 16. The pattern will continue like this. The general pattern is $2^{n+1}-1$. Thanks also to Jordi from Hall Meadow who gave the same answer. Amrit, from Newton Farm, noticed that these were both geometric series, and gave us a formula for them. (Charlie's first term was 1/2 and his ratio was 1/2; Alison's first term was 1 and her ratio was 2.) Well spotted! Niharika gave - and proved! - a general formula for $1 + a + a^2 + \dots + a^n$. Her proof went as follows: I looked at some cases when a and n were both small, and sooner or later I guessed the formula $1 + a + a^2 + \dots + a^n = \frac{1 - a^n}{\frac{1}{a} - 1}$. Now suppose I know that $1 + a + a^2 + \dots + a^k = \frac{1 - a^k}{\frac{1}{a} - 1}$ for some number k. What happens when I add $a^{k+1}$ to both sides? Well, the right hand side becomes $\frac{1 - a^k}{\frac{1}{a} - 1} + a^{k+1}$ $= \frac{1 - a^k}{\frac{1}{a} - 1} + \frac{a^{k+1}\left(\frac{1}{a} - 1\right)}{\frac{1}{a} - 1}$ $= \frac{1 - a^k}{\frac{1}{a} - 1} + \frac{a^{k} - a^{k-1}}{\frac{1}{a} - 1}$ $= \frac{1 - a^{k+1}}{\frac{1}{a} - 1}$. What does this mean? It means that, if I know $1 + a + a^2 + \dots + a^k = \frac{1 - a^k}{\frac{1}{a} - 1}$ is true for some certain number k, then I know $1 + a + a^2 + \dots + a^{k+1} = \frac{1 - a^{k+1}}{\frac{1}{a} - 1}$ is true for k+1 too. So if I know that I'm allowed to plug in n = k into the formula $1 + a + a^2 + \dots + a^n = \frac{1 - a^n}{\frac{1}{a} - 1}$, then I know I'm also allowed to plug in n = k+1. But I can check it works for n = 1, so it must work for n = 2; but if it works for n = 2, it must work for n = 3; but if it works for n = 3, it must work for n = 4, and so on. It works for all values of n! Wow! Thanks, Niharika. (For anyone who's interested, this is a special type of argument called a "proof by induction".) Niharika also found a method of using Charlie's sum to find the answer to Alison's sum, which she thought was 'cute': Let's start off with Alison's sum, but without the 1 at the start: $2 + 4 + 8 + \dots + 2^n$ $= 2^{n+1}\left( \frac{1}{2^n} + \frac{1}{2^{n-1}} + \dots + \frac{1}{8} + \frac{1}{4} + \frac{1}{2}\right)$ and the bit in brackets is Charlie's sum, which we've already worked out, so we get: $= 2^{n+1} \left(\frac{2^n - 1}{2^n}\right)$ $= 2^{n+1} - 2$, and then we just add the 1 back to both sides. That is cute. Well done!</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><h3>Why do this problem</h3> In this problem a surprising number pattern can be explained by using an image. By understanding the significance of the image offered students are helped to perceive the general rule. <h3> </h3> <h3>Possible approach</h3> "Imagine a sequence of fractions where each one is half of the previous fraction." Write these sums on the board and ask students to work them out: <div> $$\frac{1}{2} + \frac{1}{4} $$ $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8}$$ $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} +\frac{1}{16}$$ "What do you notice?" "Do you think the pattern will continue?" "How do you know?" Offer students a chance to share their ideas, and then show the video, or recreate Charlie's diagram on the board. Perhaps ask students to recreate the diagram for themselves. "How could you use the diagram to explain the patterns you have noticed?" "Can you describe an expression for the sum $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} +\dots + \frac{1}{2^n}$?" "Can you convince someone that your expression is correct for all values of $n$?" </div> <div> </div> <div>Allow students some time to discuss in pairs, then bring the class together to share their insights. It is important to insist on clearly justified arguments that refer to the generality - a key question to ask is "How do you know it will always happen?".</div> <div> </div> <div>The second part of the problem, looking at the sum of the sequence $1 + 2 + 4 + \dots + 2^n$, can be treated in the same way.</div> <div> </div> <h3>Possible extension</h3> <a href="/6700">Diminishing Returns</a> uses similar diagrams to explore other geometric series and can lead on to discussion of infinite sums. <h3> Possible support</h3> Students could start by working on <a href="/6710">Slick Summing</a>, in which they are invited to explore the sums of simple arithmetic sequences. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Watch the videos carefully. For the first video, what happens to the area remaining as Charlie adds on each new fraction? For the second video, what is the relationship between the new area added, and the total area?</mdoxml> 2 3 0 0 0 1 0 Double Trouble Simple additions can lead to intriguing results... Numbers and the Number System Powers & roots Sequences, Functions and Graphs Sequences Advanced Algebra Summation of series Using, Applying and Reasoning about Mathematics Visualising