Sigma n3

Q.

$\frac{1^3}{1}+\frac{(1^3+2^3)}{(1+2)}+\frac{(1^3+2^3+3^3)}{(1+2+3)}+....15terms=$

1. $446$

2. $680$

3. $600$

4. $540$

Q. The sum of the series ${31}^3+{32}^3+\dots \dots +{50}^3$ is

1. $98092$
2. $121023$
3. $1409400$
4. $150569$

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.

If ${\sum }_{r-1}^{n}r=55$, find ${\sum }_{r-1}^n{r}^3$

Q. Find the sum of series to $n$ terms:
$1\cdot 2\cdot 5+2\cdot 3\cdot 6+3\cdot 4\cdot 7+.....$

Q. The sum to $n$ terms of the series $(1\times 2\times 3)+(2\times 3\times 4)+(3\times 4\times 5)+\dots$ is

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

Q. The value of $\underset{n\to \infty }{lim}\sum _{k=0}^n\frac{{}^n{C}_K}{n^K(K+3)}$ is equal to

1. $e$
2. $e-2$
3. $e-1$
4. $1-e$

Q.

FInd the $n^{th}$ term of the series $(1{)}^3$ + ($1^3$ + $2^3$) + ($1^3$ + $2^3$ + $3^3$) + ------------------- n terms

1. 

2. 

3. 

4. 

Q.

$\frac{9}{1!}+\frac{19}{2!}+\frac{35}{3!}+\frac{57}{4!}+....+\infty$

1. $12e$

2. $12e-4$

3. $12e-3$

4. $12e-5$

Q. The sum of the series $1^3-2^3+3^3-\dots \dots +9^3=?$

1. $125$
2. $300$
3. $425$
4. $0$

Q. Find the sum of the series : 1.n+2. (n-1) + 3.(n-2)+....+(n-1).2+n.1.

Q. Let n be a fixed positive integer. Defined a relation $R$ on the set $Z$ of integers by, $aRb$ $\leftrightarrow$ $n|a-b$. Then what is the relation $R$

1. reflexive
2. symmetric
3. transitive
4. equivalence
5. None of the above

Q. If $y=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+.......$ find $\frac{dy}{dx}$

Q. The value of $\underset{n\to \infty }{lim}\frac{\sum _{i=1}^n(i{)}^{p+q+1}}{\sum _{i=1}^n(i{)}^q\sum _{i=1}^n(i{)}^p}$ is equal to (where $p,q\ge 1$)

1. $\frac{pq}{p+q+2}$
2. $\frac{(p+1)(q+1)}{p+q+2}$
3. $\frac{(p-1)(q-1)}{p+q+1}$
4. $\frac{(pq{)}^2}{p+q+1}$

Q. Find the sum of the series $S=1^2-2^2+3^2-4^2+...+{29}^2-{30}^2$ ?

1. $-454$
2. $-464$
3. $-455$
4. $-465$

Q. The sum to $n$ terms of the series $(1\times 2\times 3)+(2\times 3\times 4)+(3\times 4\times 5)+\dots$ is

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

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 $n^{th}$ term of the $r^{th}$ order of figurate numbers is equal to the $(n+2{)}^{th}$ term of the $(r-2{)}^{th}$ order$,$ show that $r=n+2$.

Q. If $f\left(n\right)={[(n+1)(n+2)...(n+n)]}^{1/n}$, then ${lim}_{n\to \infty }f\left(n\right)$ equals

1. $e$
2. $1/e$
3. $2/e$
4. $4/e$

Q. The value of $\sum _{i=1}^n\sum _{j=1}^i\sum _{k=1}^{j}1=220,$ then the value of $n$ equals

1. $11$
2. $12$
3. $10$
4. $9$

Q. Let $S_n$ denote the sum of the cubes of the first $n$ natural numbers and $s_n$ denote the sum of the first $n$ natural numbers. Then ${\sum }_{r=1}^n\frac{S_r}{s_r}$ equals

1. $\frac{n(n+1)}{2}$
2. None of these
3. $\frac{n(n+1)(n+2)}{6}$
4. $\frac{n^2+3n+2}{2}$

Q. Find sum.
$3\times 8+6\times 11+9\times 14+....$

Q. The difference between the sum of the first $k$ terms of the series $1^3+2^3+3^3+\dots \dots +n^3$ and the sum of the first $k$ terms of $1+2+3+\dots \dots n$ is $1980$. The value of $k$ is

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 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. 96
2. 142
3. 192
4. 71

Q. Simplify: $\frac{23}{2^3\times 5^2}$

Q. Let the sum of the series
$(1^2+1)!+(2^2+1)2!+(3^2+1)3!+...+(n^2+1)n!$ be $n(n+k)!$. Find $k+4$ ?

Q. $\int \frac{10x^9+{10}^x{\log }_{e}10dx}{x^{10}+{10}^x}$ equals

1. ${10}^x-x^{10}+C$
2. ${10}^x+x^{10}+C$
3. $({10}^x-x^{10}{)}^{-1}+C$
4. $\log ({10}^x+x^{10})+C$

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.

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