Summation by Sigma Method

Q.

Find the sum to $n$ terms of the series $5+11+19+29+41+...$

Q.

If ${10}^{2y}=25$ then ${10}^{-y}$ equals to

1. $-\frac{1}{5}$

2. $\frac{1}{50}$

3. $\frac{1}{625}$

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

Q.

Complete the following Series $2,9,28,?,126,217,344.$

Q.

How do you evaluate ${}^{9}C_3$?

Q.

Solve:$ax+by=c,bx+ay=1+c$

Q. The sum $\frac{3\times 1^3}{1^2}+\frac{5\times (1^3+2^3)}{1^2+2^2}+\frac{7\times (1^3+2^3+3^3)}{1^2+2^2+3^2}+\cdots$ upto ${10}^{\text{th}}$ term, is :

1. $600$
2. $620$
3. $660$
4. $680$

Q.

Find the missing number.

$120,80,40,45,?,5$

Q. Detailed difference between sequence, series and Progression.

Q.

If the sum of $n$ terms of an A.P is$2n^2+5n$, then the $nth$ term will be:

1. $4n+3$

2. $4n+5$

3. $4n+6$

4. $4n+7$

Q.

The number of real solutions of ${\tan }^{-1}\left(\sqrt{x}\left(x-1\right)\right)+{\sin }^{-1}\left(\sqrt{x^2+x+1}\right)=\frac{\pi }{2}$ is

1. $0$

2. $1$

3. $2$

4. $\infty$

Q.

The value of $\left({}^{21}{C}_1-{}^{10}C_1\right)+\left({}^{21}{C}_2-{}^{10}{C}_2\right)+\left({}^{21}{C}_3-{}^{10}{C}_3\right)+\left({}^{21}{C}_4-{}^{10}{C}_4\right)+......+\left({}^{21}{C}_{10}-{}^{10}{C}_{10}\right)$ is

1. $2^{21}-2^{10}$

2. $2^{20}-2^9$

3. $2^{20}-2^{10}$

4. $2^{21}-2^{11}$

Q. The sum of the series $1+2\times 3+3\times 5+4\times 7+\dots \text{upto}{11}^{\text{th}}\text{term}$ is :

1. $916$
2. $945$
3. $946$
4. $915$

Q.

Write square numbers and cube numbers from $1$ to $30$ natural numbers.

Q.

Find the sum of $2+4+6+....+200$.

Q.

Find three consecutive even numbers whose sum is $96$.

Q.

The equation $e^{\sin x}-e^{-\sin x}-4=0$ has

1. infinite number of real roots

2. no real roots

3. exactly one real root

4. exactly four roots

Q.

Solve the solution: $1+4+7+10..........................+x=287$

Q.

Write the following number in expanded form :

$343876792$

Q. Total number of prime numbers between $90$ and $110$ is

Q.

Find three consecutive even numbers whose sum is $234$.

Q. If $\left(1\right)\left(2020\right)+\left(2\right)\left(2019\right)+\left(3\right)\left(2018\right)+\cdots +\left(2020\right)\left(1\right)=2020\times 2021\times k,$ then the value of $k$ is

1. $337$
2. $2019$
3. $365$
4. $\frac{2019}{2}$

Q.

If the sum of first $n$ terms of an AP is $c{n}^2$, then the sum of squares of these $n$ terms is

1. $\raisebox{1ex}{$n(4n^2-1)c^2$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

2. $\raisebox{1ex}{$n(4n^2+1)c^2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

3. $\raisebox{1ex}{$n(4n^2-1)c^2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

4. None of these

Q. If $\int \frac{2e^x+3e^{-x}}{4e^x+7e^{-x}}dx=\frac{1}{14}(ux+v{\log }_e(4e^x+7e^{-x}\left)\right)+C$, where $C$ is a constant of integration, then $u+v$ is equal to

Q.

If ${\tan }^{-1}\left(\frac{x-1}{x+1}\right)+{\tan }^{-1}\left(\frac{2x-1}{2x+1}\right)={\tan }^{-1}\left(\frac{23}{36}\right)$, then $x=$

1. $\frac{3}{4},-\frac{3}{8}$

2. $\frac{3}{4},\frac{3}{8}$

3. $\frac{4}{3},\frac{3}{8}$

4. None of these

Q.

If every term of a $G.P$ with positive terms is the sum of its two previous terms, then common ratio of the series is

Q.

Find the sum of the given series $45+46+47+...+113+114+115$

Q.

Solve the following simultaneous equation using Cramers rule.

$$\begin{gathered} 3x-4y=10 \\ 4x+3y=5 \end{gathered}$$

Q.

If $a_1,a_2,...,a_n,$ are in AP with a common difference $d\ne 0$, then the sum of the following series is$(\sin d)[\cos ec{a}_1\cos ec{a}_2+\cos ec{a}_2\cos ec{a}_3+...+\cos ec{a}_{(n-1)}\cos ec{a}_n]$ is equal to

1. $sec{a}_1-sec{a}_n$

2. $cot{a}_1-cot{a}_n$

3. $\tan a_1-\tan a_n$

4. $\cos ec{a}_1-\cos ec{a}_n$

Q.

The sum of all the products of the first $n$ natural numbers taken two at a time is

1. $\frac{1}{24}n\left(n-1\right)\left(n+1\right)\left(3n+2\right)$

2. $\frac{n^2}{48}\left(n-1\right)\left(n-2\right)$

3. $\frac{1}{6}n\left(n+1\right)\left(n+2\right)\left(n+5\right)$

4. None of these

Q.

The solution of the equation $1+\mathrm{a}+{\mathrm{a}}^2+{\mathrm{a}}^3+...{\mathrm{a}}^{\mathrm{x}}=(1+\mathrm{a})(1+{\mathrm{a}}^2)(1+{\mathrm{a}}^4)$. Then $\mathrm{x}$ is equal to

1. $3$

2. $5$

3. $7$

4. None of these