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Find the coefficient of x^(n-1) in the expansion of [ 1+2x+3x^2......+ nx^(n-1) ]^2.?
Please i tried a lot to solve this problem. Any hint/suggestion/answer would be REALLY appreciated.
Also my book says that the coefficient is equal to 1•n +2(n-1) + 3(n-2) +4(n-3)..... (n-1)•2 + n•1
Since i've already found out this value, just explain me WHY the coefficient is equal to this. BA to be given.
1 Answer
- AmyLv 75 years ago
The expansion of that square will be the sum of all terms that can be made by multiplying any term of the polynomial with any term of the polynomial. For example, one term in the sum is 2x * nx^(n-1).
You are only interested in the terms that will end up having degree (n-1).
For example, the x^5 term multiplied by the x^(n-6) term.
Those are: 1 * nx^(n-1), 2x * (n-1)x^(n-2), 3x^2 * (n-2)x^(n-3), etc. all the way to nx^(n-1) * 1.
Their sum is x^(n-1) * (1*n + 2*(n-1) + 3*(n-2) ... + n*1)