Coefficient of xn-1 in the expansion of, (x+3)n+(x+3)n-1(x+2)+(x+3)n-2(x+2)2+ ldots . .+(x+2)n is :

Question

Coefficient of xn-1 in the expansion of, (x+3)n+(x+3)n-1(x+2)+(x+3)n-2(x+2)2+ ldots . .+(x+2)n is : (A)  n+1 C2(3) (B)  n-1 C2(5) (C)  n+1 C2(5) (D)  n C2(5)

Tardigrade Admin · Accepted Answer

(x+3)n+(x+3)n-1(x+2)+ ldots ldots .+(x+2)n=(x+3)n((1-((x+2/x+3))n+1/1-(x+2)x+3)) =[(x+3)n+1-(x+2)n+1] Coefficient of xn-1= n+1 Cn-1(3)2- n+1 Cn-1 × 4

Q. Coefficient of $x^{n - 1}$ in the expansion of, $(x + 3)^{n} + (x + 3)^{n - 1} (x + 2) + (x + 3)^{n - 2} (x + 2)^{2} + \dots .. + (x + 2)^{n}$ is :

$^{n + 1} C_{2} (3)$

$^{n - 1} C_{2} (5)$

$^{n + 1} C_{2} (5)$

$^{n} C_{2} (5)$

$(x + 3)^{n} + (x + 3)^{n - 1} (x + 2) + \dots\dots . + (x + 2)^{n} = (x + 3)^{n} (\frac{1 - ( \frac{x + 2}{x + 3} ) ^{n + 1}}{1 - \frac{x + 2}{x + 3}})$
$= [(x + 3)^{n + 1} - (x + 2)^{n + 1}]$
Coefficient of $x^{n - 1} =^{n + 1} C_{n - 1} (3)^{2} -^{n + 1} C_{n - 1} \times 4$

Q. Coefficient of xn−1 in the expansion of, (x+3)n+(x+3)n−1(x+2)+(x+3)n−2(x+2)2+…..+(x+2)n is :

n+1C2​(3)

n−1C2​(5)

n+1C2​(5)

nC2​(5)