Search any question & find its solution
Question:
Answered & Verified by Expert
If the coefficients of $x^4, x^5$ and $x^6$ in the expansion of $(1+x)^n$ are in the arithmetic progression, then the maximum value of $n$ is:
Options:
Solution:
2492 Upvotes
Verified Answer
The correct answer is:
14
$\begin{aligned} & \text { Coeff. of } x^4={ }^n C_4 \\ & \text { Coeff. of } x^5={ }^n C_5 \\ & \text { Coeff. of } x^6={ }^n C_6 \\ & { }^n C_4,{ }^n C_5,{ }^n C_6 \ldots . A P \\ & 2 C_5={ }^n C_4+{ }^n C_6 \\ & 2=\frac{{ }^n C_4}{{ }^n C_5}+\frac{{ }^n C_6}{{ }^n C_5} \quad\left\{\frac{{ }^n C_r}{{ }^n C_r}=\frac{n-r+1}{r}\right\} \\ & 2=\frac{5}{n-4}+\frac{n-5}{6} \\ & 12(n-4)=30+n^2-9 n+20 \\ & n^2-21 n+98=0 \\ & (n-14)(n-7)=0 \\ & n_{\max }=14 \quad n_{\min }=7\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.