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  • If the coefficients of <img alt="p^{\text {th }},(p+1)^{\text {th }} \text { and }(p+2)^{\text {th }}" src="https://entrancecorner.oncodecogs.com/gif.latex?p%5E%7B%5Ctext%20%7Bth%20%7D%7D%2C%2

If the coefficients of p^{\text {th }},(p+1)^{\text {th }} \text { and }(p+2)^{\text {th }} terms in the expansion of (1+x)^n are in A.P., then

Option: 1

n^2-2 n p+4 p^2=0


Option: 2

n^2-n(4 p+1)+4 p^2-2=0


Option: 3

n^2-n(4 p+1)+4 p^2=0


Option: 4

None of these


Answers (1)

best_answer

Coefficient of p^{\text {th }},(p+1)^{\text {th }} \text { and }(p+2)^{\text {th }} terms in expansion of (1+x)^n are { }^n C_{p-1},{ }^n C_\rho,{ }^n C_{p+1}

\\Then\ 2^n C_p={ }^n C_{p-1}+{ }^n C_{p+1}\\ \Rightarrow n^2-n(4 p+1)+4 p^2-2=0

Trick : Let p  = 1, hence  { }^n C_0,{ }^n C_1 \text { and }{ }^n C_2 are in A.P

\begin{aligned} & \Rightarrow 2 \cdot{ }^n C_1={ }^n C_0+{ }^n C_2 \Rightarrow 2 n=1+\frac{n(n-1)}{2} \\ & \Rightarrow 4 n=2+n^2-n \Rightarrow n^2-5 n+2=0 \end{aligned}

which is given by (b).

 

 

Posted by

Divya Prakash Singh

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