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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Sequences and Series

NDA 2023 · Mathematics · Sequences and Series

If G is the geometric mean of numbers 1,2,2^2, 2^3, , 2 n^ -1 ,...

If G is the geometric mean of numbers 1,2,22,23,…,2n−11,2,2^2, 2^3, \ldots, 2 n^{-1}, then what is the value of 1+2log⁡2G1+2 \log _2 \mathrm{G} ?

Options

  1. A11
  2. B44
  3. Cn−1n-1
  4. Dnn

Correct answer

D. nn

Step-by-step solution

1,2,22,23,……..2n−11,2,2^2, 2^3, \ldots \ldots..2 n^{-1} Number of terms =n∴GM=(1×2×22×23×……×2n−1)1/n=(21+2+3+…+(n−1))1/n=[2n(n−1)2]1/n⇒G=2n−12∴1+2log⁡2G=1+2log⁡22n−12=1+2(n−1)2log⁡22=1+n−1=n=n\therefore \quad \mathrm{GM}=\left(1 \times 2 \times 2^2 \times 2^3 \times \ldots \ldots \times 2^{n-1}\right)^{1 / n}=\left(2^{1+2+3+\ldots+(n-1)}\right)^{1 / n}=\left[2^{\frac{n(n-1)}{2}}\right]^{1 / n} \Rightarrow G=2^{\frac{n-1}{2}}\therefore \quad 1+2 \log _2 G=1+2 \log _2 2^{\frac{n-1}{2}}=1+2 \frac{(n-1)}{2} \log _2 2=1+n-1=n
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