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Evaluate the sum using generating functions.

Evaluate the sum (using generating functions)

A) 0+3+12+...+3n2.
B) 4x3x2x1+5x4x3x2+...+n(n-1)(n-2)(n-3)

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uestion

Evaluate the sum (using generating functions)
A) 0+3+12+...+3n2.
B) 4x3x2x1+5x4x3x2+...+n(n-1)(n-2)(n-3)

A) Solution:

Here we have to find the sum of 0+3+12+...+3n^2

This can be written as 3(0+1^2+2^2+...+n^2)

(ie) 3(1^2+2^2+...+n^2)

3 summation Sn , where sn = 1^2+2^2+...+n^2

First we will find the sum of 1^2+2^2+...+n^2

Here sn = 1^2+2^2+...+n^2

We consider the identity

k^3- (k-1)^3 = k^3 - (k^3-3k^2+3k-1) [ (a-b)^3= a^3-3a^2b+3ab^2-b^3)

= k^3- k^3 +3k^2-3k+1

k^3-(k-1)^3 = 3k^2-3k+1

Putting k= 1,2,3...n successively, we obtain

1^3 - 0^3 = 3(1)^2 - 3(1) +1

2^3-1^3 = 3(2)^2-3(2)+1

3^3-2^3 = 3(3)^2 ...

Solution Summary

Sums are evaluated using generating functions.

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