Q152 / 270 · NDA 2022 (Phase 2) · Mathematics

What is $\sum_{r=0}^n 2^r C(n, r)$ equal to?

  • A
    $2^n$
  • Correct answer
    B
    $3^{\mathrm{n}}$
  • C
    $2^{2 n}$
  • D
    $3^{2 n}$

Solution

$\begin{aligned} & \sum_{r=0}^n 2^r{ }^{\mathrm{n}} C r={ }^{\mathrm{n}} C_0 2^0+{ }^{\mathrm{n}} C_1 2^1+{ }^{\mathrm{n}} C_2 2^2+\ldots+{ }^{\mathrm{n}} C_2 2^{\mathrm{n}} \\ & =(1+2)^{\mathrm{n}}=3^{\mathrm{n}}\end{aligned}$

Hence, the correct answer is option: 'B'.

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