Exams › JEE Main › Maths

Evaluate the sum of binomial coefficients with indices divisible by 4: \(\binom{4n}{0} + \binom{4n}{4} + \binom{4n}{8} + \cdots + \binom{4n}{4n}\).

1. \((-1)^n 2^{2n-1} + 2^{4n-2}\)
2. \((-1)^n 2^{2n-1}\)
3. \((-1)^n 2^{4n-1}\)
4. None of these

Correct answer: \((-1)^n 2^{4n-1}\)

Solution

The sum of binomial coefficients with indices divisible by 4 can be derived using roots of unity, specifically applying the formula for the sum of binomial coefficients at specific intervals. The correct option captures the alternating sign and the exponential growth based on the number of terms, leading to the result of ((-1)^n 2^{4n-1}).

Related JEE Main Maths questions

• Let $P_n=\cos^n\theta+\sin^n\theta$. If $P_n-P_{n-2}=kP_{n-4}$, then the value of $k$ is:
• For any natural number $n$, the expression $x^{2n-1}+y^{2n-1}$ is divisible by which of the following?
• For a natural number $n$, which of the following always divides the expression $x^{n+1}+(x+1)^{2n-1}$?
• For a positive integer $n$, the expression $5^{2n+2}-24n-25$ is divisible by which of the following?
• For every natural number $n$, the expression $41^n-14^n$ is divisible by which of the following?
• Let \(C_0, C_1, C_2, \ldots, C_{15}\) denote the binomial coefficients in the expansion of \((1+x)^{15}\). Then the value of \(\frac{C_1}{C_0}+\frac{C_2}{C_1}+\frac{C_3}{C_2}+\cdots+\frac{C_{15}}{C_{14}}\) is:

⚔️ Practice JEE Main Maths free + battle 1v1 →