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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Sequences and Series

NDA 2021 · Mathematics · Sequences and Series

If g is the geometric mean of 2,4,8,16,32,64,128,256,512 , 1024,...

If gg is the geometric mean of 2,4,8,16,32,64,128,256,5122,4,8,16,32,64,128,256,512, 1024, then which one of the following is correct?

Options

  1. A8<g<168 < g < 16
  2. B16<g<3216 < g < 32
  3. C32<g<6432 < g < 64
  4. Dg>64g > 64

Correct answer

C. 32<g<6432 < g < 64

Step-by-step solution

If gg is the geometric mean of 2,4,8,16,32,64,1282,4,8,16,32,64,128, 256,512,1024256,512,1024. Then g=(2×4×8×16×32×64×128×256×512×1024)1/10g=(2 \times 4 \times 8 \times 16 \times 32 \times 64 \times 128 \times 256 \times 512 \times 1024)^{1 / 10}g=(21×22×23×24×25×26×27×28×29×210)1/10g=(21+2+3+…+10)1/10g=210(10+1)2×110⇒g=25.5∴25<g<26⇒32<g<64\begin{aligned} & g=\left(2^1 \times 2^2 \times 2^3 \times 2^4 \times 2^5 \times 2^6 \times 2^7 \times 2^8 \times 2^9 \times 2^{10}\right)^{1 / 10} \\ & g=\left(2^{1+2}+3+\ldots+10\right)^{1 / 10} \\ & g=2^{\frac{10(10+1)}{2}} \times \frac{1}{10} \Rightarrow g=2^{5.5} \\ & \therefore 2^5 < g < 2^6 \Rightarrow 32 < g < 64\end{aligned}$$
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