Method of Difference

Q.

The $n^{th}$ term of the series $2+4+7+....$ will be

1. $\frac{n^2+n+1}{2}$

2. $n^2+n+2$

3. $\frac{n^2+n+2}{2}$

4. $\frac{n^2+2n+2}{2}$

Q.

The sum of the series 1 + $\frac{1.3}{6}$ + $\frac{\mathrm{1.3.5}}{6.8}$ + ............∞ is

1. 

2. 4

3. 0

4. 1

Q.

What is the sum of infinite series?

Q.

If $n^{th}$ term of the series 1 + 5 + 12 + 22 + 35 .........can be written as $T_n$ = a$n^2$ + bn + c Find the sum of the 16 terms of the series.

1. 7076

2. 2176

3. 3000

4. 3176

Q.

If $n^{th}$ term of the series 5 + 7 + 13 + 31+ 85 ------------- can be written as $T_n$ = a.$3^{(n-1)}$ + bn + c. Find the sum of the first eight terms of the given series.

__

Q. The sum of $n$ terms of the series $1+3+7+15+31+...n$ terms is .

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

Q. If $\sum _{r=1}^n{T}_r=\frac{n}{8}(n+1)(n+2)(n+3)$, and $\sum _{r=1}^n\frac{1}{T_r}=\frac{n^2+3n}{4\sum _{k=1}^{p}k}$, then $p$ is equal to

1. $n+1$
2. $n$
3. $n-1$
4. $2n$

Q.

Find the sum of 10 terms of the series whose $n^{th}$ term is 3.$2^n$.

__

Q.

Find the $8^{th}$ term of the series 1 + 5 + 18 + 58 + 179 + ..........

__

Q.

$\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+....100$ terms $=$

1. $2^{100}+99$

2. $2^{-100}+99$

3. $2^{-101}+99$

4. $2^{-99}+99$

Q.

The sum of the series 1 + 2x + 3 $x^2$ + 4 $x^3$ + ..........upto n

terms is

1. 

2. 

3. None of these

4. 

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. The sum of n terms of series
$1^2+{2.2}^2+3^2+{2.4}^2+5^2+{2.6}^2+...$
is $\frac{n{(n+1)}^2}{2}$ when n is even. When n is odd, the sum is

1. $\frac{n^2(n+1)}{2}$
2. $\frac{n(n^2+1)}{2}$
3. $2{(n+1)}^2.(2n+1)$
4. none of these

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. Find the sum of the infinite series
$\frac{4}{1!}+\frac{11}{2!}+\frac{22}{3!}+\frac{37}{4!}+\frac{56}{5!}+....$

Q.

A person has $2$ parents, $4$grandparents, and so on. Find the numbers of his ancestors during the ten-generation preceding his own.

Q. The sum to $n$ terms of the series $\frac{1}{(1+x)(1+3x)}+\frac{1}{(1+3x)(1+5x)}+\frac{1}{(1+5x)(1+7x)}+\dots \dots$, where $n>5,x\ge 2$ is

1. $\frac{n}{(1+x)[1+(2n+1\left)x\right]}$
2. $\frac{1}{(1+x)[1+(2n+1\left)x\right]}$
3. $\frac{(1+x)}{n[1+(2n+1\left)x\right]}$
4. $\frac{1+(2n+1)x}{n(1+x)}$

Q.

The sum of infinite terms of the following series

1 + $\frac{4}{5}$ + $\frac{7}{5^2}$ + $\frac{10}{5^3}$ + ...........∞ will be

1. $\frac{3}{16}$

2. $\frac{35}{8}$

3. $\frac{35}{4}$

4. $\frac{35}{16}$

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. The $n^{th}$ term of a sequence of numbers is $a_n$ and given by the formula $a_n=a_{n-1}+2n$ for $n\ge 2$ and $a_1=1$.

Using the above information $a_n$ will be

1. $a_n=n^2+n-1$
2. $a_n=n^2+n+1$
3. $a_n=n^2-n+1$
4. $a_n=n^2-n-1$

Q.

Find the $n^{th}$ term of the series. 1 + 4 +13 +40 + 121 +.................

1. (

2. (

3. (

4. (

Q. $S=1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+...$ Find the value of 16S.

Q. Sum of the series: $(1+x)+(1+x+x^2)+(1+x+x^2+x^3)+...$ upto $n$ terms

1. $\frac{1}{1-x}[n-\frac{x^2(1-x^n)}{1-x}]$
2. $\frac{1}{1-x}[n-\frac{x^3(1-x^n)}{1-x}]$
3. $\frac{1}{1-x}[n-\frac{x(1-x^n)}{1-x}]$
4. $Noneofthese$

Q. The value of $\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+....+\frac{1}{99\times 100}$ is ____.

1. Greater than $\frac{100}{99}$
2. Equal to $\frac{100}{99}$
3. Less than $\frac{99}{100}$
4. Equal to $\frac{99}{100}$

Q. If $S_n=\frac{1}{1\times 3}+\frac{1}{3\times 5}+\frac{1}{5\times 7}+\cdots n\text{terms}$, then $S_{\infty }$ is

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

Q. The sum of the series $6+13+22+33+\dots \dots$ upto $20$ terms is

1. $3730$
2. $3720$
3. $3710$
4. $3750$

Q. $\text{Let}{a}_1=2,a_{n+1}=a_n^2-a_n+1\text{for}n\ge 1,\text{then}\sum _{n=1}^{\infty }\frac{1}{a_n}\text{is}$

Q.

The sum of 1 + $\frac{2}{5}$ + $\frac{3}{5^2}$ + $\frac{4}{5^3}$ + .............upto n terms is

1. $\frac{25}{16}$ - $\frac{4n+5}{16\times 5^{n-1}}$

2. $\frac{3}{4}$ - $\frac{2n+5}{16\times 5^{n+1}}$

3. $\frac{3}{7}$ - $\frac{3n+5}{16\times 5^{n-1}}$

4. $\frac{1}{2}$ - $\frac{5n+1}{3\times 5^{n+2}}$

Q.

The sum of the series 1 + 2x + 3 $x^2$ + 4 $x^3$ + ..........upto n

terms is

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

2. $\frac{1-x^n}{1-x}$

3. $\frac{x}{n+1}$

4. None of these

Q. If the sum of the first $15$ terms of the series $3+7+14+24+37+.......$ is $15k$, then $k$ is equal to:

1. $81$
2. $122$
3. $119$
4. $126$