Evaluate the sum using generating functions.
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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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Solution Summary
Sums are evaluated using generating functions.
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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 ...
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