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  • If the coefficients of second, third and fourth term in the expansion of are in A.P., th

If the coefficients of second, third and fourth term in the expansion of (1+x)^{2n} are in A.P., then 2 n^2-9 n+7 is equal to
 

Option: 1

-1


Option: 2

0


Option: 3

1


Option: 4

\frac{3}{2}


Answers (1)

best_answer

T_2={ }^{2 n} C_1 \times x\quad T_3={ }^{2 n} C_2 \times x^2 \quad T_4={ }^{2 n} C_3 \times x^3\\\\ \text{Coefficient of}\ T_2, T_3, T_4\ \text{are in A.P.}\\\\ \begin{aligned} & =2 \cdot{ }^{2 n} C_2={ }^{20} C_1+{ }^{2 n} C_3 \\ & =2 \frac{2 n !}{2 !(2 n-2) !}=\frac{2 n !}{(2 n-1) !}+\frac{2 n !}{3 !(2 n-3) !} \\ & =\frac{2 \cdot 2 n(2 n-1)}{2}=2 n+\frac{2 n(2 n-1)(2 n-2)}{6} \\ & =n(2 n-1)-n+\frac{(n)(2 n-1)(2 n-2)}{6} \\ & =6\left(2 n^2-n\right)=6 n+4 n^3-6 n^2+2 n \\ & =6 n(2 n-1)=2 n\left(2 n^2-3 n+4\right) \\ & =6 n-3=2 n^2-3 n+4 \\ & =0=2 n^2-9 n+7 \Rightarrow 2 n^2-9 n+7=0 . \end{aligned}

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