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I just started to read "Friendly Introduction to Number Theory". Now I'm trying exercises 1-2

1.2. Try adding up the first few odd numbers and see if the numbers you get satisfy some sort of pattern. Once you find the pattern, express it as a formula. Give a geometric verification that your formula is correct.

https://www.math.brown.edu/~jhs/frintch1ch6.pdf

I think the pattern is Σ[n=0..n](2*n + 1)

But what is geometric verification? How should I answer this question?

enter image description here

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    $\begingroup$ That's not the pattern they're looking for. Do you see the pattern in the numbers $1,4,9,16,25,36$... $\endgroup$ – Matt Samuel Oct 14 '18 at 5:12
  • $\begingroup$ @MattSamuel Hello! I can see this pattern. 1=(1^2),4=(2^2),9=(3^2),16=(4^2),25=(5^2),36=(6^2). Is that the answer they are looking for? $\endgroup$ – zono Oct 14 '18 at 5:17
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    $\begingroup$ Yes, except I have no idea what they mean by geometric verification. Presumably this is covered in the chapter, which I did not myself read. $\endgroup$ – Matt Samuel Oct 14 '18 at 5:18
  • $\begingroup$ Thank you for your answer. I understand it. $\endgroup$ – zono Oct 14 '18 at 5:20
  • $\begingroup$ You're welcome. I'd post an actual answer, but it's 1:22 am so I think I can sacrifice the rep for sleep. $\endgroup$ – Matt Samuel Oct 14 '18 at 5:22
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For one odd you get $1$ which is presented by a red square.

For tow odds you get $1+3$ which is presented by one red square and three yellow square and together they make a $2\times 2$ square.

You are supposed to find out that when you keep adding odd numbers you will get a square for the sum.

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  • $\begingroup$ Thanks! I think your answer is answering the question of geometric verification. Is that right? $\endgroup$ – zono Oct 14 '18 at 5:42
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    $\begingroup$ Yes,it is. The idea is that always a square is made out of sum of odds. $\endgroup$ – Mohammad Riazi-Kermani Oct 15 '18 at 15:43

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