Question

`(1+x+x^2+cdots+x^12)^3 = 1+cdots+ Ax^12` find A =?

Answer 1

`91`

Explanation:

`1 + x + x^2 + ... + x^12 = (1 - x^13)/(1 - x)`
`=> (1 + x + x^2 + ... + x^12)^3 = (1 - x^13)^3 / (1-x)^3`

`1/(1 - x)^3 = (1 + x + x^2 + x^3 + ... )^3`
`= 1 + C(3,1) x + C(4,2) x^2 + C(5,3) x^3 + ...`

`"So we have"`

`(1 - 3 x^13 + 3 x^26 - x^39)(1 + 3 x + 6 x^2 + 10 x^3 + ...)`

`"We need the coefficient of "x^12.`

`"This coefficient is "C(14,12) = 91.`