JAMIA 2021 — Mathematics PYQ

General Formula

For an expansion of the form $(x+a{)}^n+(x-a{)}^n$:

• If $n$ is even, the number of terms is $\frac{n}{2}+1$.

• If $n$ is odd, the number of terms is $\frac{n+1}{2}$.

Solving

In the given expression, $n=100$, which is an even number.

Using the expansion of binomial theorem:

Adding the two expansions, the terms with odd powers of $a$ (like $a^1,a^3,\dots ,a^{99}$) cancel out. The terms with even powers of $a$ (like $a^0,a^2,\dots ,a^{100}$) are doubled:

Calculation of Number of Terms

The terms are corresponding to the even subscripts $0,2,4,\dots ,100$. This is an Arithmetic Progression: $0,2,4,\dots ,100$.

Alternatively, using the formula for even $n$: