JEE Mathematics 2026 Question with Solution

Given: We need the sum of all possible values of $n \in \mathbb{N}$ such that the coefficients of $x$, $x^2$ and $x^3$ in $(1+x^2)^2(1+x)^n$ are in arithmetic progression.

Find: The required sum of values of $n$.

First expand

So we consider

Using the general term of $(1+x)^n$, the relevant coefficients are:

• Coefficient of $x$: $\binom{n}{1} = n$
• Coefficient of $x^2$: $\binom{n}{2} + 2\binom{n}{0} = \frac{n(n-1)}{2} + 2$
• Coefficient of $x^3$: $\binom{n}{3} + 2\binom{n}{1} = \frac{n(n-1)(n-2)}{6} + 2n$

Since these three coefficients are in arithmetic progression,

Therefore,

This gives

Solving the resulting cubic equation for $n \in \mathbb{N}$ yields

Therefore, the sum of all possible values of $n$ is $3$. Hence, the correct option is B.

Given: We need values of $n$ for which the coefficients of $x$, $x^2$ and $x^3$ are in arithmetic progression.

Find: The sum of all such values.

For a quick test, try small natural values of $n$. For $n=3$,

The coefficients of $x$, $x^2$ and $x^3$ become

which are in arithmetic progression with common difference $2$.

Thus $n=3$ works, and from the solution equation this is the only natural value. Therefore, the required sum is $3$ and the correct option is B.