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  • If   is a positive integer, then in the trinomial expansion of <img alt="\left(x^2+2 x+2\right)^n" src="https://entr

If  n is a positive integer, then in the trinomial expansion of \left(x^2+2 x+2\right)^n coefficient of:

Option: 1

x is n.2^n


Option: 2

x^2 \text { is } n^2 \cdot\left(2^{n-1}-1\right)


Option: 3

x^3 \text { is } 2^n \cdot{ }^n C_3


Option: 4

All of these


Answers (1)

best_answer

\begin{aligned} & \left(x^2+2 x+2\right)^n=(2+x(2+x))^n \\ & =2^n+{ }^n C_1 2^{n-1} x(2+x)+{ }^n C_2 2^{n-2} x^2(2+x)^2+{ }^n C_3 2^{n-3} x^3(2+x)^3+\ldots \ldots \end{aligned}

It is easy to observe that coefficient of x=n.2^n

Coefficient of x^2=n \cdot 2^{n-1}+\frac{n(n-1) 2^{n-2}}{2}=n^2 \cdot 2^{n-1}

Coefficient of x^3={ }^n C_2 \cdot 2^{n-2} \cdot 4+{ }^n C_3 \cdot 2^{n-3} \cdot 2^3

The last one can be simplified to 2^n \cdot{ }^{n+1} C_3.

So the only correct option is (a)

Posted by

Rishi

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