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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Binomial Theorem

NDA 2023 · Mathematics · Binomial Theorem

Let (1+x)^n=1+T_1 x+T_2 x^2+T_3 x^3+ +T_n x^n What is T_1+T_2+T_3+...

Let (1+x)n=1+T1x+T2x2+T3x3+…+Tnxn(1+x)^n=1+T_1 x+T_2 x^2+T_3 x^3+\ldots+T_n x^n What is T1+T2+T3+…+TnT_1+T_2+T_3+\ldots+T_n equal to ?

Options

  1. A2n2^{\mathrm{n}}
  2. B2n−12^{\mathrm{n}}-1
  3. C2n−12^{\mathrm{n}-1}
  4. D2n+12^{\mathrm{n}}+1

Correct answer

B. 2n−12^{\mathrm{n}}-1

Step-by-step solution

Putting x=1\mathrm{x}=1 in given statement (1+1)n=1+T1+T2(1+1)^{\mathrm{n}}=1+\mathrm{T}_1+\mathrm{T}_2+T3+…+Tn⇒T1+T2+T3+….+Tn=2n−1\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}$$
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