what-is-coefficient-of-x-2-in-1-x-1-2x-1-3x-1-4x-1-5x-1-14x-1-15x-

Question Number 177822 by cortano1 last updated on 09/Oct/22

what is coefficient of x^{2} in

(1 - x) (1 + 2 x) (1 - 3 x) (1 + 4 x) (1 - 5 x)

\dots (1 + 14 x) (1 - 15 x) ?

Commented by mr W last updated on 10/Oct/22

d o y o u h a v e t h e s o l u t i o n ?

Commented by cortano1 last updated on 11/Oct/22

coefficient of x^{2} in the expansion

of the product \underset{k = 1}{\prod^{n}} (1 + a_{k} x)

is equal to \sum_{i \neq j} a_{i} a_{j}

\Rightarrow \sum_{i \neq j} a_{i} a_{j} = \frac{1}{2} (\sum_{i, j} a_{i} a_{j} - \sum_{j} a_{j}^{2})

= \frac{1}{2} ({(\sum_{j} a_{j})}^{2} - \sum_{j} a_{j}^{2})

= \frac{1}{2} ({(- 8)}^{2} - 1240)

= - 588

Answered by mr W last updated on 09/Oct/22

{(1 + 3 + 5 + \dots + 15)}^{2} / 2 + {(2 + 4 + 6 + \dots + 14)}^{2} / 2

- (1^{2} + 2^{2} + 3^{2} + \dots + 15^{2}) / 2

- (1 + 3 + 5 + \dots + 15) (2 + 4 + 6 + \dots + 14)

= 64^{2} / 2 + 56^{2} / 2 - \frac{15 \times 16 \times (2 \times 15 + 1)}{12} - 64 \times 56

= - 588

Commented by Tawa11 last updated on 10/Oct/22

Great sir

Commented by cortano1 last updated on 11/Oct/22

yes - 588