Question

If the coefficients of $x^4$, $x^5$, and $x^6$ in the expansion of $(1 + x)^n$ are in arithmetic progression, then the maximum value of $n$ is:

• A. 14
• B. 21
• C. 28
• D. 7

Answer 1

Answer: (A)

14

Answer 2

In the binomial expansion of $(1+x)^n$, the general term is given by $T_k = \binom{n}{k}x^k$. Therefore, the coefficients of $x^4$, $x^5$, and $x^6$ are:

• Coefficient of $x^4$: $\binom{n}{4}$,
• Coefficient of $x^5$: $\binom{n}{5}$,
• Coefficient of $x^6$: $\binom{n}{6}$.

Since these coefficients are in an arithmetic progression, we can set up the condition:

$2\binom{n}{5} = \binom{n}{4} + \binom{n}{6}.$

Using the formula for binomial coefficients, we have:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}.$

After simplifying, we substitute and solve for $n$ to find that the maximum value of $n$ that satisfies this condition is $n = 14$.

Therefore, the maximum value of $n$ is $14$.