Binomial Theorem
Coefficients in AP — Sum of Valid n
nta_pyq_2026_jan
Grade 11
Question:
The sum of all possible values of $n\in\mathbf{N}$, so that the coefficients of $x$, $x^2$ and $x^3$ in the expansion of $(1+x^2)^2(1+x)^n$, are in arithmetic progression is:
Step-by-Step Solution
Key Concept: $(1+x^2)^2=1+2x^2+x^4$. Coefficients in $(1+x^2)^2(1+x)^n$: coeff of $x$: $\binom{n}{1}=n$; coeff of $x^2$: $\binom{n}{2}+2$; coeff of $x^3$: $\binom{n}{3}+2n$.
$n=2,3,4$. Sum $=9$.
Correct Answer: 4