The greatest value of n for which 1 - 1-2 - 1-2-2 -1-2-n - 2 is -n -N-

The explanation for the correct option.Find the greatest value.Given 1+1/2+1/2^2+.....+1/2^n Sum of GP =a(1 r^n)/1 r = 1(1 (1/2)^n+1)/(1 1/2)0ex0ex= 2(1 1/2 ...

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Standard XII

Mathematics

Theorems for Differentiability

The greatest ...

Question

The greatest value of $n$ for which $1 + \frac{1}{2} + \frac{1}{2^{2}} + . . . . . + \frac{1}{2^{n}} < 2$ is $(n \in N)$ .

$100$

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$10$

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$1000$

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None of these

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Solution

The correct option is D
None of these

The explanation for the correct option.
Find the greatest value.
Given $1 + \frac{1}{2} + \frac{1}{2^{2}} + . . . . . + \frac{1}{2^{n}} < 2$
Sum of GP $= \frac{a (1 - r^{n})}{1 - r}$
$= \frac{1 (1 - {(\frac{1}{2})}^{n + 1})}{(1 - \frac{1}{2})} = 2 (1 - \frac{1}{2^{n + 1}}) = 2 - \frac{1}{2^{n}} < 2$
Hence, option $D$ is the correct answer.

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The greatest value of n for which 1+12+122+.....+12n<2 is (n∈N).

The correct option is D None of theseThe explanation for the correct option.Find the greatest value.Given 1+12+122+.....+12n<2Sum of GP =a(1-rn)1-r=11-12n+11-12=21-12n+1=2-12n<2Hence, option D is the correct answer.

The greatest value of $n$ for which $1 + \frac{1}{2} + \frac{1}{2^{2}} + . . . . . + \frac{1}{2^{n}} < 2$ is $(n \in N)$ .