Binomial Theorem
Binomial Theorem
nta_pyq_2025_jan
Grade 11
Question:
For some $n\ne 10$, let the coefficients of the $5^{\text{th}}$, $6^{\text{th}}$ and $7^{\text{th}}$ terms in the binomial expansion of $(1+x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1+x)^{n+4}$ is:
Step-by-Step Solution
Key Concept: The $k^{\text{th}}$ term's coefficient in $(1+x)^{m}$ is $\binom{m}{k-1}$. A.P.\ condition becomes $2\binom{m}{5}=\binom{m}{4}+\binom{m}{6}$, a quadratic in $m=n+4.$
Let $m=n+4.$ A.P.: $2\binom{m}{5}=\binom{m}{4}+\binom{m}{6}.$ Divide by $\binom{m}{5}$:
$$2=\frac{5}{m-4}+\frac{m-5}{6}.$$
Multiply by $6(m-4)$: $12(m-4)=30+(m-5)(m-4)\Rightarrow m^{2}-21m+98=0\Rightarrow m=7\text{ or }14.$
$m=14\Rightarrow n=10$ (rejected). So $m=7,\,n=3$.
Largest coefficient in $(1+x)^{7}$ is $\binom{7}{3}=\binom{7}{4}=35.$
Correct Answer: 3