Sigma n3

Q.

The value of sum of the series: $3·{}^{n}C_0-8·{}^{n}C_1+13·{}^{n}C_2-18·{}^{n}C_3+......$up to $\left(n+1\right)$ terms is

1. $0$

2. $3n$

3. $5n$

4. None of these

Q. The sum of first 9 terms of the series.
$\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+...$

1. 142
2. 192
3. 71
4. 96

Q.

Find the sum of the squares of the first n natural numbers.

1. $\frac{n^2(n+1)}{6}$

2. $\frac{n(n+1)(n-1)}{6}$

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

4. $\frac{n(n+1)(2n+1)}{6}$

Q.

Find the sum of the cubes of the first $n$ natural numbers.

1. $n^2(n+1{)}^2$

2. $\left(n\right)(n+1)(2n+1)$

3. $[\frac{n(n+1)}{2}{]}^2$

4. None of these

Q.

If $\frac{(1+i)x-2i}{3+i}+\frac{(2-3i)y+i}{3-i}=i$ , then the real values of x and y are given by :

1. x = -3, y= - 1

2. x= 3, y= - 1

3. x=3, y=1

4. x=1, y=-3

Q. Expand using Binomial Theorem ${(1+\frac{x}{2}-\frac{2}{x})}^4,x\ne 0$ and let the sum of coefficients of the terms in the expansion be $t$. Find $10000t$

Q. Let $A_1=\left[a_1\right]A_2=\left[\begin{array}{cc}{a}_2& a_3\\ a_4& a_5\end{array}\right]A_3=\left[\begin{array}{ccc}{a}_6& a_7& a_8\\ a_9& a_{10}& a_{11}\\ a_{12}& a_{13}& a_{14}\end{array}\right]\cdots \cdots A_n=[\cdots ]\text{where}{a}_r=\left[{\log }_{2}r\right]$
$\left(\right[\cdot ]$ denotes the greatest integer$)$. Then trace of $A_{10}$ is

Q. Using Binomial Theorem, evaluate $(101{)}^4$

Q.

Find the sum of the series s = 1 + $\frac{1}{2}$(1 + 2) + $\frac{1}{3}$(1 + 2 + 3) + $\frac{1}{4}$(1 + 2 + 3 + 4) + .........upto 40 terms.

__

Q.

Find the sum of $1.n^2+2(n-1{)}^2+3(n-2{)}^2+......n{.1}^2$

1. 

2. 

3. 

4. 

Q.

The sum of (n+1) terms of $\frac{1}{2}+\frac{1}{1+2}+\frac{1}{1+2+3}+..........$ is

1. 

2. 

3. 

4. 

Q. Find $(a+b{)}^4-(a-b{)}^4$. Hence find $s$ if $(\sqrt{3}+\sqrt{2}{)}^4-(\sqrt{3}-\sqrt{2}{)}^4=40\sqrt{s}$

Q.

The sum of the series $1^2$.2 + $2^2$.3 + $3^2$.4 + ........ to n terms is

1. 

2. 

3. 

4. 

Q.

The sum of n terms of three A.P.s whose first term is 1 and

common differences are 1, 2, 3 are $S_1$, $S_2$, $S_3$ respectively. The true

relation is

1. 

2. 

3. 

4. 

Q.

The sum of (n+1) terms of $\frac{1}{1}+\frac{1}{1+2}+\frac{1}{1+2+3}+..........$ is

1. 

2. 

3. 

4. 

Q. If the sum of the first $15$ terms of the series
${\left(\frac{3}{4}\right)}^3+{\left(1\frac{1}{2}\right)}^3+{\left(2\frac{1}{4}\right)}^3+3^3+{\left(3\frac{3}{4}\right)}^3+....$

is equal to $225k$, then $k$ is equal to :

1. $9$
2. $27$
3. $54$
4. $108$

Q.

If the sum of the $3^3$ + $7^3$ + ${11}^3$ + ${15}^3$ ...................20 terms is $s_{20}$. Find the value of $\frac{s_{20}}{100}$

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Q.

If $\frac{3+5+7+......nterms}{5+8+11+.........+10terms}$ = 7, the value of n is:

1. 35

2. 36

3. 37

4. 40

Q. If $3k,$ $k,$ $[k^2-34]$ are the first three terms of a geometric progression, where $k\in R^{+}$ and $[.]$ is the greatest integer function, then the value of $\sum _{r=1}^{10}(r{)}^{k/2}$ is

1. $55$
2. $385$
3. $3025$
4. $325$

Q. The value of the determinant of $n^{th}$ order, given by
$\left|\begin{array}{cccc}x& 1& 1& \cdots \\ 1& x& 1& \cdots \\ 1& 1& x& \cdots \\ \cdot & \cdot & \cdot & \cdots \end{array}\right|$ is

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