in the expansion of (3 - 2x) (1 + x/2) ^n, the coefficient of x is 7. find the value of the constant n and hence find the coefficient of x^2

n = 6

Coefficient of x^2 = 21/4.

Step-by-step explanation:

(3 - 2x) (1 + x/2) ^n

= 3 (1 + x/2) ^n - 2x (1 + x/2) ^n

= 3 (1 + n*1 * (x/2) + nC2 * (x/2) ^2 + ...) - 2x (1 + n (x/2) + nC2 (x/2) ^2 + ...)

= 3 + 3n (x/2) + 3 nC2 (x/2) ^2 + ...) - 2x - 2xn (x/2) ...)

So 3n (x/2) - 2x = 7x

3n/2 - 2 = 7

3n/2 = 9

n = 6

Coefficient of x^2 = 3 * 6C2 (1/2) ^2 - 2*6 * 1/2

= 3 * 15 * 1/4 - 6

= 45/4 - 6

= 21/4.