Sum of First n Odd Numbers

Date: 7/10/96 at 8:7:2
From: Wong Cheong Siong
Subject: Sum of First n Odd Numbers

Dear Dr. Math,

Would you please help me answer one mathematics question that has 
puzzled me for quite some time? This question is as follows:

Show that the sum of the first n odd numbers is a perfect square. 
Show also that 57^2 - 13^2 is the sum of certain consecutive odd 
numbers, and find them.

Date: 7/10/96 at 10:18:20
From: Doctor Ethan
Subject: Re: Sum of First n Odd Numbers

Okay, let's look at the sum in question. It is
1 +3 + 5 + ...... + 2n-1   so there are n terms.

We can rewrite it like this

2n-1 + 2n-3 + 2n-5 + ...
1    +    3 +    5 + ...

Now if we add these up we have n/2 terms of value 2n. (You can figure 
out why this doesn't mess up when n is odd.)

So the total for the sum is  2n * n/2 = n^2.

Now on to question 2.

If the first n odds add to n^2 then the first 57 odds add to 57^2
and the first 13 odds add to 13^2. So, which odds add to 57^2 - 13^2?

Good luck.

-Doctor Ethan,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 7/11/96 at 6:45:50
From: Wong Cheong Siong
Subject: Re: Sum of First n Odd Numbers

Dear Dr . Math,

I do not quite understand the solution you have just sent me.
Especially the part where you said if we add these up we have n/2 
terms of value 2n. Would you please send me a concise explanation of 
each step? I definitely appreciate your help very much.

Date: 7/11/96 at 10:39:34
From: Doctor Ethan
Subject: Re: Sum of First n Odd Numbers

Okay I'll try again.

>Okay let's look at the sum in question: it is
>1 +3 + 5 + ...... + 2n-1

So there are n terms. Rewrite it like this:
2n-1 + 2n-3 + 2n-5 + ...  This list is n/2 terms long.
1    +    3 +    5 + ...  This list is n/2 terms long.

Now we add vertically 2n -1 + 1 = 2n
2n -3 +3 =2n.
2n -5 + 5= 2n.

Now if we add these up we have n/2 terms of value 2n. (You can figure
out why this doesn't mess up when n is odd.)

2n + 2n + 2n + .....  This has n/2 terms.

So the total for the sum is  2n * n/2 = n^2.

-Doctor Ethan,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 7/11/96 at 10:26:40
From: Doctor Jerry
Subject: Re: Sum of First n Odd Numbers

Sometimes a second method of solving a problem helps.  Here's one
possibility.  

Let

     S=1+3+5+...+(2n-1)      
           Add 2+4+...+2n to both sides.
     S+2+4+...+2n=1+2+3+...+2n.    
           On the left,
     S+2+4+...+2n=S+2(1+2+...+n)=S+2n(n+1)/2=S+n^2+n 
           On the right,
     1+2+3+...+2n=2n(1+2n)/2=n+2n^2.  So,
     S+n^2+n=n+2n^2.                  So,
     S=n^2.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/