Find the coefficient of $x^{48}$ in the product $(x - 2) (x - 4) (x - 6) . . . . . (x - 98)$ is

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Solution

$(x - 2) (x - 4) (x - 6) . . . . . (x - 98)$ has $\frac{98}{2} = 49$ terms

$∴ (x - 2) (x - 4) (x - 6) . . . . . (x - 98) = x^{49} + a_{1} x^{48} + a_{2} x^{47} + a_{3} x^{46} + . . . . . . + a_{48} x + a_{49}$

Coefficient of $x^{48} = a_{1}$

Since the roots are $2, 4, 6, 8, . . .98$

$a_{1} = - (2 + 4 + . . . + 96 + 98)$

$= - 2 (1 + 2 + 3 + . . . + 48 + 49)$

$= - 2 \frac{49 (49 + 1)}{2}$ since sum of natural numbers $= \frac{n (n + 1)}{2}$

$= - 49 \times 50 = - 2450$

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