IIT-JEE 1994Question 1
Let $n$ be a positive integer. If the coefficients of the 2nd, 3rd and 4th terms in the expansion of $(1+x)^{n}$ are in A.P., then the value of $n$ is ______.
Binomial Theorem · Properties and Sums of Binomial Coefficients
Answer$7$
Solution
Concept. The coefficients of the 2nd, 3rd, 4th terms are ${}^{n}C_1,{}^{n}C_2,{}^{n}C_3$. Formulas used. (1) $p,q,s$ in A.P. $\iff2q=p+s$Step 1 - Condition. By (1), $2\,{}^{n}C_2={}^{n}C_1+{}^{n}C_3$: $$n(n-1)=n+\frac{n(n-1)(n-2)}{6}.$$ Step 2 - Simplify. Divide by $n$ and multiply by $6$: $$6n-12=n^2-3n+2\ \Longrightarrow\ n^2-9n+14=0.$$ So $n=2$ or $n=7$. Step 3 - Choose. A 4th term needs $n\ge3$, so $n=7$. Answer: 7