Method of Difference

Q. The sum $\sum _{k=1}^{20}k\frac{1}{2^k}$ is equal to :

1. $2-\frac{11}{2^{19}}$
2. $2-\frac{21}{2^{20}}$
3. $1-\frac{11}{2^{20}}$
4. $2-\frac{3}{2^{17}}$

Q. Find the sum of odd integers from $1$ to $2001$.

Q.

${\displaystyle \sum _{r=1}^{10}}r!\left(r^3+6r^2+2r+5\right)=\alpha \left(11!\right)$. What is the value of $\alpha$?

Q. The sum $1+\frac{1^3+2^3}{1+2}+\frac{1^3+2^3+3^3}{1+2+3}+\cdot \cdot \cdot +\frac{1^3+2^3+3^3+\cdot \cdot \cdot +{15}^3}{1+2+3+\cdot \cdot \cdot +15}-\frac{1}{2}(1+2+3+\cdot \cdot \cdot +15)$ is equal to :

1. $1240$
2. $620$
3. $1860$
4. $660$

Q.

If $a_n=\sum _{n=0}^{\infty }\frac{1}{{}^{n}C_r}$then $\sum _{n=0}^{\infty }\frac{r}{{}^{n}C_r}$equals:

1. $\left(n-1\right)a_n$

2. $n{a}_n$

3. $\frac{n}{2}{a}_n$

4. none of these

Q. Sum of the first $20$ terms of the series
$1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+\cdots$

1. $38+\frac{1}{2^{19}}$
2. $40+\frac{1}{2^{19}}$
3. $38+\frac{1}{2^{18}}$
4. $40+\frac{1}{2^{18}}$

Q. Find the sum of the series:
$\left(i\right)2+5+8+..+182$

Q. Let the $n$ terms of series is $\frac{1}{\mathrm{1.2.3.4}}+\frac{1}{\mathrm{2.3.4.5}}+\frac{1}{\mathrm{3.4.5.6}}+\dots \dots$ then

1. $T_n=\frac{1}{n(n+1)(n+2)(n+3)}$
2. $S_{\infty }=\frac{1}{6}$
3. $S_{\infty }=\frac{1}{18}$
4. $S_n=\frac{1}{3(n+1)(n+2)(n+3)}$

Q.

If the third term of a G.P is $4$ then the product of its first $5$ terms is

1. $4^3$

2. $4^4$

3. $4^5$

4. None of these

Q.

Sum of infinite series $1+\frac{2}{3}·\frac{1}{2}+\frac{2}{3}·\frac{5}{6}·\frac{1}{2^2}+\frac{2}{3}·\frac{5}{6}·\frac{8}{9}·\frac{1}{2^3}+...$ is:

1. $2^{\frac{1}{3}}$

2. $4^{\frac{1}{3}}$

3. $8^{\frac{1}{3}}$

4. $2^{\frac{1}{5}}$

Q. Let $S=\frac{1^2}{1.3}+\frac{2^2}{3.5}+\frac{3^2}{5.7}+......+\frac{{500}^2}{999.1001}$ then the number of positive divisors of $\left[S\right]$ is
(where $[.]$ denotes the greatest integer function)

Q. There is a certain sequence of positive real numbers. Beginning from the third term, each term of the sequence is the sum of all the previous terms. The seventh term is equal to 1000 and the first term is equal to 1. The second term of this sequence is equal to

1. 
2. 
3. 
4. 

Q. If the sum of $n$ terms of the series $5+7+13+31+85+\dots$ is $\frac{1}{2}(a^n+bn+c)$, then

1. $b+5c=a$
2. $b+4c=a$
3. $\frac{b}{a+c}=4$
4. $\frac{b}{a+c}=2$

Q. The sum of the infinite series $1+\frac{2}{3}+\frac{7}{3^2}+\frac{12}{3^3}+\frac{17}{3^4}+\frac{22}{3^5}+\dots$ is equal to :

1. $\frac{9}{4}$
2. $\frac{15}{4}$
3. $\frac{13}{4}$
4. $\frac{11}{4}$

Q. The sum of the $n$ terms of the series $3+8+22+72+266+1036+\cdots$

1. $n(n+1)+\frac{1}{3}(4^n-1)$
2. $n(n-1)+\frac{1}{3}(4^n-1)$
3. $n(n+3)+\frac{1}{3}(4^n-1)$
4. $n(n-2)+\frac{1}{3}(4^n-1)$

Q.

The first term of an infinite geometric progression is x and its sum is 5. Then

1. 

2. 

3. 

4. 

Q. Sum of first $n$ terms of the sequence $5,7,11,17,25,\dots$ is equal to

1. $\frac{n}{6}(2n^2+15)$
2. $\frac{n}{3}(n^2+14)$
3. $n(2n+3)$
4. $\frac{n}{6}(n^2+14)$

Q.

If ${\mathrm{S}}_{\mathrm{n}}=\frac{1}{1^3}+\frac{1+2}{1^3+2^3}+\frac{1+2+3}{1^3+2^3+3^3}+\dots ..+\frac{1+2+\dots \dots +\mathrm{n}}{1^3+2^3+\dots \dots +{\mathrm{n}}^3};n=1,2,3..$ Then $S_n$ is not greater than:

1. $\frac{1}{2}$

2. $1$

3. $2$

4. $4$

Q. If $S_n,T_n$ represents the sum of $n$ terms and $n^{\text{th}}$ term respectively of the series $1+4+10+20+35+\cdots$, then $\frac{n{T}_{n+1}}{S_n}=$

Q.

If the sum of the first $40$ terms of the series, $3+4+8+9+13+14+18+19+\dots ..$ is $\left(102\right)m$, then $m$ is equal to:

1. $10$

2. $25$

3. $5$

4. $20$

Q. Let $g\left(x\right)$ be a continuous function satisfying $g(x+a)+g\left(x\right)=0,\forall x\in R,a>0.$ If $\underset{b}{\overset{2k}{\int }}g\left(t\right)dt$ is independent of $b$ and $b,k,c$ are in A.P. , then the least positive value of $c$ is

1. $\sqrt{a}$
2. $a$
3. $a^2$
4. $2a$

Q. If $S_2$ and $S_4$ denotes respectively the sum of the squares and the sum of the fourth powers of first $n$ natural number, then $\frac{S_4}{S_2}=$ ________.

Q. If odd natural numbers are arranged in groups as $\left(1\right),(3,5),(7,9,11),...$ Then the sum of the numbers in the ${10}^{\text{th}}$ group is

1. $500$
2. $729$
3. $743$
4. $1000$

Q. $\underset{n\to \infty }{lim}\frac{n(2n+1{)}^2}{(n+2)(n^2+3n-1)}$ is equal to

1. $0$
2. $2$
3. $4$
4. $\infty$

Q. Sum of the series $\sum _{r=1}^n\frac{1}{(ar+b)(ar+a+b)}$, (where $a\ne 0$) is

1. $\frac{n}{(a+b)\left[a\right(n+1)+b]}$
2. $\frac{n}{(a+b)[an+b]}$
3. $\frac{n}{(a+b)\left[a\right(n-1)+b]}$
4. $\frac{n}{(a+b)\left[a\right(n+2)+b]}$

Q. If the $n^{\text{th}}$ term and the sum of $n$ terms of the series $2,12,36,80,150,252,.....$ is $T_n$ and $S_n$ repectively then

1. $T_n=n^3+n^2$
2. $\frac{1}{12}\left(n\right)(n+1)(n+2)(3n+2)$
3. $T_n=n^3-n^2$
4. $\frac{1}{12}\left(n\right)(n+1)(n+2)(3n+1)$

Q. How many words, with or without meaning can be made form the letter of the word MONDAY, assuming that no letter is repeated , if. all letters are used at a time.

Q.

Each term of the sum
$C_0^2+3.C_1^2+5.C_2^2.....(2n+1)c_n^2$ can be written as $(2r+1){}^n{C}_r^2$

1. True

2. False

Q. The sum to $(n+1)$ terms of the series $\frac{C_0}{2}-\frac{C_1}{3}+\frac{C_2}{4}-\frac{C_3}{5}+\cdots$ is

1. $\frac{1}{n+1}$
2. $\frac{1}{n+2}$
3. $\frac{1}{n(n+1)}$
4. $\frac{1}{(n+1)(n+2)}$

Q. It is known that ${\sum }_{r=1}^{\infty }\frac{1}{{(2r-1)}^2}=\frac{{\pi }^2}{8}$. Then ${\sum }_{r=1}^{\infty }\frac{1}{r^2}$ is equal to

1. none of these
2. $\frac{{\pi }^2}{24}$
3. $\frac{{\pi }^2}{3}$
4. $\frac{{\pi }^2}{6}$