The sum of the first n terms of the series (1)/(2)+(3)/(4)+(7)/(8)+(15)/(16)+ . is equal to

Correct answer: C. 1-2^ -n aligned & (1)/(2)+(3)/(4)+(7)/(8)+(15)/(16)+ .. & =(1-(1)/(2))+(1-(1)/(4))+(1-(1)/(8))+(1-(1)/(16))+ & =(1+1+1+ . n)-((1)/(2)+(1)/(4)+(1)/(8)+(1)/(16)+ ..) & =n-((1)/(2)+(1)/(2^2)+(1)/(2^3)+(1)/(2^4)+ ..) & =n- (1)/(2)(1- 1 2^ n ) 1-(1)/(2) & =n- (1)/(2)(1- 1 2^ n ) (1)/(2) & =n-1+2^ -n & =2^ -n +n-1 aligned

NDA 2017

Exam: NDA
Year: 2017
Subject: Mathematics
Chapter: Sequences and Series

NDA 2017 · Mathematics · Sequences and Series

The sum of the first n terms of the series...

The sum of the first n terms of the series \frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\ldots$. is equal to

Options

A $2^{\mathrm{n}}-\mathrm{n}-1$
B $1-2^{-n}$
C $1-2^{-n}$
D $2^{\mathrm{n}}-1$

Correct answer

C. $1-2^{-n}$

Step-by-step solution

\begin{aligned} & \frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\ldots.. \\ & =\left(1-\frac{1}{2}\right)+\left(1-\frac{1}{4}\right)+\left(1-\frac{1}{8}\right)+\left(1-\frac{1}{16}\right)+\ldots \ldots \\ & =(1+1+1+\ldots. \mathrm{n})-\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\ldots..\right) \\ & =\mathrm{n}-\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\ldots..\right) \\ & =\mathrm{n}-\frac{\frac{1}{2}\left(1-\frac{1}{2^{\mathrm{n}}}\right)}{1-\frac{1}{2}} \\ & =\mathrm{n}-\frac{\frac{1}{2}\left(1-\frac{1}{2^{\mathrm{n}}}\right)}{\frac{1}{2}} \\ & =\mathrm{n}-1+2^{-\mathrm{n}} \\ & =2^{-\mathrm{n}}+\mathrm{n}-1\end{aligned}

Practice Sequences and Series on Quantrex Academy →

More from Sequences and Series

Let p , q and r be three unequal numbers such that p , q and r are in AP. If (q-p)… NDA 2026 If p^ th term of an AP is k , then what is the sum of p^ th term, (p+q)^ th term and… NDA 2026 If p , g_1 , g_2 and q are in GP and m is the arithmetic mean of p and q , then… NDA 2026 The sum of the first n terms of an AP is 3n^2+5n . If the m^ th term of the AP is 68… NDA 2026 Passage: Let X(a,p) , Y(b,q) and Z(c,r) be the points such that a , b and c are in AP.… NDA 2026 If (10 + _ 10 x) , (10 + _ 10 y) and (10 + _ 10 z) are in AP, then consider the following… NDA 2025 How many terms of the series 1 + 3 + 5 + 7 + … amount to a sum equal to 12345678987654321… NDA 2025 For the following two (02) items: Let (6 + 10 + 14 + … up to m terms) = (1 + 3 + 5 + 7 +… NDA 2025 For the following two (02) items: Let (6 + 10 + 14 + … up to m terms) = (1 + 3 + 5 + 7 +… NDA 2025 How many terms are identical in the two APs 19, 21, 23, … up to 110 terms and 19, 22, 25… NDA 2025 If p , 1 , q are in AP and p , 2 , q are in GP, then which of the following statements… NDA 2025 The sum of the first 8 terms of a GP is five times the sum of its first 4 terms. If r ≠ 1… NDA 2025 If 5th, 7th and 13th terms of an AP are in GP, then what is the ratio of its first term… NDA 2025 The sum of the first k terms of a series S is 3k^2 + 5k . Which one of the following is… NDA 2025 What is the harmonic mean of the numbers C(10,3) , C(10,4) , C(10,5) , C(10,6) and… NDA 2025 If p times the p^ th term of an AP is equal to q times the q^ th term (p ≠ q) , then what… NDA 2024 Full Sequences and Series list Sequences and Series JEE Maths PYQs JEE Mathematics PYQs All NDA PYQs