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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 1-T_1+2 T_2-3...

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 1−T1+2T2−3T3+…+(−1)nnTn1-T_1+2 T_2-3 T_3+\ldots+(-1)^n n T_n equal to?

Options

  1. A0
  2. B−2n−1-2^{\mathrm{n}-1}
  3. Cn2n−1\mathrm{n} 2^{\mathrm{n}-1}
  4. D1

Correct answer

D. 1

Step-by-step solution

Putting, x=−1x=-1 in equation (i) in privious problem, we get n(1−1)n−1=T1−2T+3T3−4T4+……….(−1)n−1Tn⇒1−T1+2T2−3T3+4T4+…+(−1)nTn=0+1=1\begin{aligned} & \mathrm{n}(1-1)^{\mathrm{n}-1}=\mathrm{T}_1-2 \mathrm{ T}+3 \mathrm{ T}_3-4 \mathrm{ T}_4+\ldots \ldots \ldots .(-1)^{\mathrm{n}-1} \mathrm{ T}_{\mathrm{n}} \\ & \Rightarrow 1-\mathrm{T}_1+2 \mathrm{ T}_2-3 \mathrm{ T}_3+4 \mathrm{ T}_4+\ldots+(-1)^{\mathrm{n}} \mathrm{ T}_{\mathrm{n}}=0+1=1 \end{aligned}
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