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  • If the coefficients of  <img alt="\small 5^{\text {th }}, 6^{\text {th }} \text { and } 7^{\text {th }}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Csmall%205%5E%7B%5Ctext%20%

If the coefficients of  \small 5^{\text {th }}, 6^{\text {th }} \text { and } 7^{\text {th }}  terms in the expansion of \small (1+x)^n be in A.P., then n =
 

Option: 1

7 only


Option: 2

14 only
 


Option: 3

7 or 14


Option: 4

None of these
 


Answers (1)

best_answer

\text { Coefficient of } T_5={ }^n C_4, T_6={ }^n C_5 \text { and } T_7={ }^n C_6

\text { According to the condition, } 2{ }^n C_5={ }^n C_4+{ }^n C_6

\begin{aligned} & \Rightarrow 2\left[\frac{n !}{(n-5) ! 5 !}\right]=\left[\frac{n !}{(n-4) ! 4 !}+\frac{n !}{(n-6) ! 6 !}\right] \\ & \Rightarrow 2\left[\frac{1}{(n-5) 5}\right]=\left[\frac{1}{(n-4)(n-5)}+\frac{1}{6 \times 5}\right] \end{aligned}

\text { After solving, we get } n=7 \text { or } 14 \text {. }

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