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.


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

  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ 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

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.

  • $\begingroup$ Thanks! I think your answer is answering the question of geometric verification. Is that right? $\endgroup$ – zono Oct 14 '18 at 5:42
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.