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(b) Compute the sum

\ [  (a + (  2n+1  ) d  ) ^2- (a + (2n  ) d  ) ^2 + (  a + (2n-1  ) d  ) ^2 - (a+ (  2n-2  ) d  ) ^2 + \cdots + (a+d  ) ^2 - a^2.\] 

Please explain the solution to the problem, thank you. 

 Dec 6, 2019
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\ [  (a + (  2n+1  ) d  ) ^2- (a + (2n  ) d  ) ^2 + (  a + (2n-1  ) d  ) ^2 - (a+ (  2n-2  ) d  ) ^2 + \cdots + (a+d  ) ^2 - a^2.\] 

 

\(\ [ (a + ( 2n+1 ) d ) ^2- (a + (2n ) d ) ^2 + ( a + (2n-1 ) d ) ^2 - (a+ ( 2n-2 ) d ) ^2 + \cdots + (a+d ) ^2 - a^2] \)

 

I have not finished this question but i have made a reasonable start.

 

I expanded each of these squared terms.

The a^2  terms all cancel out.

So you are left with 2 series.

   One of those series looks easy to deal with, the other looks a lot more tricky.

 

Have you tried this and made a good start for yourself ?

 Dec 6, 2019

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