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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

Respuesta :

Answer:

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.

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