Q190 / 270 · NDA 2023 (Phase 2) · Mathematics

Let $(1+x)^n=1+T_1 x+T_2 x^2+T_3 x^3+\ldots+T_n x^n$ What is $T_1+T_2+T_3+\ldots+T_n$ equal to ?

  • A
    $2^{\mathrm{n}}$
  • Correct answer
    B
    $2^{\mathrm{n}}-1$
  • C
    $2^{\mathrm{n}-1}$
  • D
    $2^{\mathrm{n}}+1$

Solution

Putting $\mathrm{x}=1$ in given statement $(1+1)^{\mathrm{n}}=1+\mathrm{T}_1+\mathrm{T}_2$ $\begin{aligned} & +\mathrm{T}_3+\ldots+\mathrm{T}_{\mathrm{n}} \\ & \Rightarrow \mathrm{~T}_1+\mathrm{T}_2+\mathrm{T}_3+\ldots .+\mathrm{T}_{\mathrm{n}}=2^{\mathrm{n}}-1 \end{aligned}$

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

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