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Recent questions tagged summation

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1 answer

1

Counting number of pairs whose sum is less than k
How many pairs $(x,y)$ such that $x+y <= k$, where x y and k are integers and $x,y>=0, k > 0$. Solve by summation rules. Solve by combinatorial argument.

dd asked in Combinatory Jun 8, 2020

by dd

883 views

2 votes

2 answers

2

ISI2014-DCG-16
The sum of the series $\dfrac{1}{1.2} + \dfrac{1}{2.3}+ \cdots + \dfrac{1}{n(n+1)} + \cdots $ is $1$ $1/2$ $0$ non-existent

Arjun asked in Quantitative Aptitude Sep 23, 2019

by Arjun

410 views

1 vote

1 answer

3

ISI2014-DCG-34
The following sum of $n+1$ terms $2 + 3 \times \begin{pmatrix} n \\ 1 \end{pmatrix} + 5 \times \begin{pmatrix} n \\ 2 \end{pmatrix} + 9 \times \begin{pmatrix} n \\ 3 \end{pmatrix} + 17 \times \begin{pmatrix} n \\ 4 \end{pmatrix} + \cdots$ up to $n+1$ terms is equal to $3^{n+1}+2^{n+1}$ $3^n \times 2^n$ $3^n + 2^n$ $2 \times 3^n$

Arjun asked in Combinatory Sep 23, 2019

by Arjun

363 views

1 vote

0 answers

4

ISI2014-DCG-65
The sum $\dfrac{n}{n^2}+\dfrac{n}{n^2+1^2}+\dfrac{n}{n^2+2^2}+ \cdots + \dfrac{n}{n^2+(n-1)^2} + \cdots \cdots$ is $\frac{\pi}{4}$ $\frac{\pi}{8}$ $\frac{\pi}{6}$ $2 \pi$

Arjun asked in Quantitative Aptitude Sep 23, 2019

by Arjun

327 views

1 vote

1 answer

5

ISI2014-DCG-72
The sum $\sum_{k=1}^n (-1)^k \:\: {}^nC_k \sum_{j=0}^k (-1)^j \: \: {}^kC_j$ is equal to $-1$ $0$ $1$ $2^n$

Arjun asked in Combinatory Sep 23, 2019

by Arjun

367 views

2 votes

1 answer

6

ISI2015-MMA-17
Let $X=\frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001}$. Then, $X \lt1$ $X\gt3/2$ $1\lt X\lt 3/2$ none of the above holds

Arjun asked in Quantitative Aptitude Sep 23, 2019

by Arjun

396 views

0 votes

2 answers

7

ISI2015-MMA-24
The series $\sum_{k=2}^{\infty} \frac{1}{k(k-1)}$ converges to $-1$ $1$ $0$ does not converge

Arjun asked in Quantitative Aptitude Sep 23, 2019

by Arjun

373 views

1 vote

1 answer

8

ISI2015-MMA-54
If $0 <x<1$, then the sum of the infinite series $\frac{1}{2}x^2+\frac{2}{3}x^3+\frac{3}{4}x^4+ \cdots$ is $\log \frac{1+x}{1-x}$ $\frac{x}{1-x} + \log(1+x)$ $\frac{1}{1-x} + \log(1-x)$ $\frac{x}{1-x} + \log(1-x)$

Arjun asked in Others Sep 23, 2019

by Arjun

469 views

0 votes

0 answers

9

ISI2015-MMA-80
Let $0 < \alpha < \beta < 1$. Then $ \Sigma_{k=1}^{\infty} \int_{1/(k+\beta)}^{1/(k+\alpha)} \frac{1}{1+x} dx$ is equal to $\log_e \frac{\beta}{\alpha}$ $\log_e \frac{1+ \beta}{1 + \alpha}$ $\log_e \frac{1+\alpha }{1+ \beta}$ $\infty$

Arjun asked in Calculus Sep 23, 2019

by Arjun

309 views

1 vote

1 answer

10

ISI2015-MMA-84
For positive real numbers $a_1, a_2, \cdots, a_{100}$, let $p=\sum_{i=1}^{100} a_i \text{ and } q=\sum_{1 \leq i < j \leq 100} a_ia_j.$ Then $q=\frac{p^2}{2}$ $q^2 \geq \frac{p^2}{2}$ $q< \frac{p^2}{2}$ none of the above

Arjun asked in Others Sep 23, 2019

by Arjun

258 views

1 vote

4 answers

11

ISI2015-DCG-2
Let $S=\{6, 10, 7, 13, 5, 12, 8, 11, 9\}$ and $a=\underset{x \in S}{\Sigma} (x-9)^2$ & $b = \underset{x \in S}{\Sigma} (x-10)^2$. Then $a <b$ $a>b$ $a=b$ None of these

gatecse asked in Quantitative Aptitude Sep 18, 2019

338 views

0 votes

1 answer

12

ISI2015-DCG-15
The smallest integer $n$ for which $1+2+2^2+2^3+2^4+ \cdots +2^n$ exceeds $9999$, given that $\log_{10} 2=0.30103$, is $12$ $13$ $14$ None of these

gatecse asked in Quantitative Aptitude Sep 18, 2019

183 views

1 vote

2 answers

13

ISI2016-DCG-2
Let $S=\{6,10,7,13,5,12,8,11,9\},$ and $a=\sum_{x\in S}(x-9)^{2}\:\&\: b=\sum_{x\in S}(x-10)^{2}.$ Then $a<b$ $a>b$ $a=b$ None of these

gatecse asked in Quantitative Aptitude Sep 18, 2019

327 views

0 votes

1 answer

14

ISI2016-DCG-17
The smallest integer $n$ for which $1+2+2^{2}+2^{3}+2^{4}+\cdots+2^{n}$ exceeds $9999$, given that $\log_{10}2=0.30103$, is $12$ $13$ $14$ None of these

gatecse asked in Quantitative Aptitude Sep 18, 2019

175 views

0 votes

1 answer

15

ISI2016-DCG-23
The value of $\log_{2}e-\log_{4}e+\log_{8}e-\log_{16}e+\log_{32}e-\cdots\:\:$ is $-1$ $0$ $1$ None of these

gatecse asked in Quantitative Aptitude Sep 18, 2019

218 views

1 vote

2 answers

16

ISI2017-DCG-1
The value of $\dfrac{1}{\log_2 n}+ \dfrac{1}{\log_3 n}+\dfrac{1}{\log_4 n}+ \dots + \dfrac{1}{\log_{2017} n}\:\:($ where $n=2017!)$ is $1$ $2$ $2017$ none of these

gatecse asked in Quantitative Aptitude Sep 18, 2019

339 views

0 votes

1 answer

17

ISI2017-DCG-13
The value of $\dfrac{x}{1-x^2} + \dfrac{x^2}{1-x^4} + \dfrac{x^4}{1-x^8} + \dfrac{x^8}{1-x^{16}}$ is $\frac{1}{1-x^{16}}$ $\frac{1}{1-x^{12}}$ $\frac{1}{1-x} – \frac{1}{1-x^{16}}$ $\frac{1}{1-x} – \frac{1}{1-x^{12}}$

gatecse asked in Quantitative Aptitude Sep 18, 2019

232 views

0 votes

1 answer

18

ISI2018-DCG-27
$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}$ is $2$ $1$ $\infty$ not a convergent series

gatecse asked in Quantitative Aptitude Sep 18, 2019

213 views

2 votes

1 answer

19

Kenneth Rosen Edition 6th Exercise 2.4 Question 15c (Page No. 161)
$\sum_{j=2}^{8}(-3)^j$

aditi19 asked in Set Theory & Algebra Dec 5, 2018

287 views

3 votes

1 answer

20

ISI2016-MMA-18
Let $A=\begin{pmatrix} -1 & 2 \\ 0 & -1 \end{pmatrix}$, and $B=A+A^2+A^3+ \dots +A^{50}$. Then $B^2 =1$ $B^2 =0$ $B^2 =A$ $B^2 =B$

go_editor asked in Linear Algebra Sep 13, 2018

199 views

0 votes

0 answers

21

ISI2016-MMA-22
The infinite series $\Sigma_{n=1}^{\infty} \frac{a^n \log n}{n^2}$ converges if and only if $a \in [-1, 1)$ $a \in (-1, 1]$ $a \in [-1, 1]$ $a \in (-\infty, \infty)$

go_editor asked in Others Sep 13, 2018

186 views

2 votes

1 answer

22

Infinite series
Find the infinite sum of the series $1 + \frac{4}{7} + \frac{9}{7^2} + \frac{16}{7^3} + \frac{25}{7^4} + .............\Join$

pankaj_vir asked in Quantitative Aptitude Aug 8, 2018

764 views

0 votes

0 answers

23

Bounding Summation
How does the below bounds to logn? Please explain the steps 1 and 2. I came to know that they are using the idea of splitting the summations and bounding them. How the first and second step came?

Ayush Upadhyaya asked in Algorithms May 11, 2018

by Ayush Upadhyaya

731 views

1 vote

1 answer

24

addition
value of 1/3 + 1/15 + 1/35 +............................+1/9999 a)100/101 b)50/101 c)100/51 d)50/51

A_i_$_h asked in Quantitative Aptitude Sep 12, 2017

1.2k views

5 votes

1 answer

25

Series Summation
Series summation of $S_n$ in closed form? $\begin{align*} &S_n = \frac{1}{1.2.3.4} + \frac{1}{2.3.4.5} + \frac{1}{3.4.5.6} + \dots + \frac{1}{n.(n+1).(n+2).(n+3)} \end{align*}$

dd asked in Set Theory & Algebra Jun 11, 2017

by dd

555 views

3 votes

2 answers

26

Manipulation of sum
Prove the identity: $\begin{align*} &\sum_{i=0}^{n}\sum_{j=0}^{i} a_ia_j = \frac{1}{2}\left ( \left ( \sum_{i=0}^{n}a_i \right )^2 + \left ( \sum_{i=0}^{n}a_i^2 \right )\right ) \end{align*}$

dd asked in Combinatory Feb 25, 2017

by dd

550 views

2 votes

0 answers

27

summation series
what is the summation of this series? S=nC0*20+nC1*21+nC2*22+..............nCn*2n

firki lama asked in Combinatory Jan 17, 2017

338 views

5 votes

4 answers

28

Time complexity and output
#include <stdio.h> #define N 3 int main() { int array[N] = {1,2,3}; int i,j; for ( i=1; i<(1<<N); i++) { for( j=0; j<N; j++) { if((1<<j)&i) { printf("%d", array[j]); } } printf("\n"); } return 0 ... $N = n \;\; , n \; \text{ is a positive integer }$ ? B. What is the output? C. What will be the complexity when $N$ is large.

dd asked in Programming Dec 17, 2016

by dd

1.3k views

0 votes

1 answer

29

summation
5 digit numbers are possible from digits 1, 2, 3, 4, 5, 6, 7 When each digit is distinct is 7P5 . what is sum of all such numbers?

Neal Caffery asked in Combinatory Dec 11, 2016

by Neal Caffery

285 views

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