Sum of First n Odd NumbersDate: 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/ |
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