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

The Correct Option is A

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 \( 14 \).