Sigma n

Q. A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k, then k - 20 is equal to___

Q. If $(x^2+x+1)+(x^2+2x+3)+(x^2+3x+5)+\cdots +(x^2+20x+39)=4500,$ then $x$ is equal to

1. $10$
2. $-10$
3. $20.5$
4. $-20.5$

Q.

Evaluate the product $95\times 96$ without multiplying directly

Q. The sum of all $3-$digit numbers less than or equal to $500,$ that are formed without using the digit ${}^{\prime \prime }{1}^{\prime \prime }$ and they all are multiple of $11,$ is

Q. Let ${\left\{a_n\right\}}_{n=1}^{\infty }$ be a sequence such that $a_1=1,a_2=1$ and $a_{n+2}=2a_{n+1}+a_n$ for all $n\ge 1.$ Then the value of $47\sum _{n=1}^{\infty }\frac{a_n}{2^{3n}}$ is equal to

Q. Let $S_k,k=1,2,...,100$, denote the sum of the infinite geometric series whose first term is $\frac{k-1}{k!}$ and the common ratio is $\frac{1}{k}$. Then the value of $\frac{{100}^2}{100!}+{\sum }_{k=1}^{100}\left|\right(k^2-3k+1\left)S_k\right|$ is

Q.

The sum of first $n$ natural number is

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

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

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

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

Q.

$1+3+7+15+31...nterms=$

1. $2^{n+1}–n$

2. $2^{n+1}–n–2$

3. $2^n–n–2$

4. $noneofthese$

Q. If $S_1,S_2,S_3\dots \dots ,S_n,\dots$ are the sums of infinite geometric series whose first terms are $1,2,3,\dots \dots n,\dots \dots$ and whose common ratios are $\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots \dots \frac{1}{n+1}\dots \dots$ respectively, then the value of $\sum _{r=1}^{2n-1}{S}_r^2$ is

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

Q.

Let $S$ be the sum of the first $9$ terms of the series:

$\left\{x+ka\right\}+\left\{x^2+(k+2)a+x^3+(k+4)a+x^4+(k+6)a\right\}+\dots$ where $a\ne 0$ and $a\ne 1$.

If $S=\frac{(x^{10}-x+45a(x-1\left)\right)}{(x-1)}$, then $k$ is equal to:

1. $3$

2. $-3$

3. $1$

4. $5$

Q. The value of $1^2-2^2+3^2-4^2+5^2-6^2+\dots n\text{terms}$ is/are

1. $\frac{n}{2},$ when $n$ is odd
2. $-\frac{(n+1)}{2},$ when $n$ is even
3. $-\frac{n(n+1)}{2},$ when $n$ is even
4. $\frac{n(n+1)}{2},$ when $n$ is odd

Q.

$\frac{3}{1!}+\frac{5}{2!}+\frac{9}{3!}+\frac{15}{4!}+\frac{23}{5!}+.....\infty is$

1. $4e-1$

2. $4e-3$

3. $3e+2$

4. $3e+4$

Q. The sum of the series $\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{420}$ is

1. $\frac{19}{20}$
2. $\frac{20}{21}$
3. $\frac{20}{19}$
4. $\frac{21}{20}$

Q. If $\left(1\right)\left(2003\right)+\left(2\right)\left(2002\right)+\left(3\right)\left(2001\right)+\cdots +\left(2003\right)\left(1\right)=\left(2003\right)\left(334\right)\left(x\right),$ then the value of $\left[\frac{x}{5}\right]$ is equal to
([.] represents the greatest integer function)

Q.

Find the sum of first n natural numbers.

Q. Find the sum of first $n$ terms and the sum of first $5$ terms of the geometric series $1+\frac{2}{3}+\frac{4}{9}+\cdots$

Q.

If $\left(4^3\right)\left(4^6\right)\left(4^{12}\right).......\left(4^{3x}\right)=(0.0625{)}^{-54}$, the value of x is

1. 8

2. 9

3. 10

4. 7

Q. $(1+x{)}^n-nx-1$ is divisible by $\left(wherenϵN\right)$

1. All of these
2. 
3. 
4. 

Q.

How do you know if an infinite geometric series converges $?$

Q. Given that n numbers of A.Ms are inserted between two sets of numbers $a,2b$ and $2a,b$ where $a,b\in R.$
Suppose further that the $m^{th}$ means between these sets of numbers are same, then the ratio $a:b$ equals

1. $n–m+1:m$
2. $n–m+1:n$
3. $n:n–m+1$
4. $m:n–m+1$

Q. $n^{th}$ term of the series $1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+.....$ will be

1. 
2. 
3. 
4. 

Q. If $2^{10}+2^9\cdot 3^1+2^8\cdot 3^2+\cdots +2\cdot 3^9+3^{10}=S-2^{11},$ then $S$ is equal to:

1. $3^{11}$
   
2. $\frac{3^{11}}{2}+2^{10}$
   
3. $2\cdot 3^{11}$
   
4. $3^{11}-2^{12}$
   

Q. Sum of the series 0.5 + 0.55 + 0.555 + . . . . . . . . . upto n terms is

1. 
2. 
3. 
4. 

Q. Evaluate sum of $n$ terms of the series $\frac{8}{5}+\frac{16}{65}+\frac{24}{325}+.....$

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

Q.

The sum of the series 1+(1+2)+(1+2+3)+...........upto n terms, will be

1. 

2. 

3. 

4. 

Q.

If $\sum _{i=1}^{\pi }i$ = $\frac{n(n+1)}{2}$, then $\sum _{i=1}^{\pi }(3i-2)$ =

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

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

3. n(3n + 2)

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

Q. Sum the series $n.1+(n-1).2+(n-2).3+....1.n.$ is $\frac{n(n+1)(n+2)}{6}$.If you think this is true write 1 otherwise write 0

Q. The sum of first 'n' terms of the series $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+....is$

1. 
2. 
3. 
4. 

Q.

The sum of the series ${20}_{c_0}-{20}_{c_1}+{20}_{c_2}-{20}_{c_3}+...{20}_{c_{10}}is$

1. 

2. 

3. 

4. 

Q.

A sequence is defined recursively by the given formulas. Find the first five terms of the sequence.

$a_n=2(a_{n-1}+2)and{a}_1=3$