Sum of n Terms

Q. Let $a_1,a_2,\cdots ,a_{4001}$ are in A.P.. If $\frac{1}{a_1{a}_2}+\frac{1}{a_2{a}_3}+\cdots +\frac{1}{a_{4000}{a}_{4001}}=10$ and $a_2+a_{4000}=50,$ then

1. $a_1{a}_{4001}=400$
2. $a_1{a}_{4001}=401$
3. $|a_1-a_{4001}|=40$
4. $|a_1-a_{4001}|=30$

Q.

Find the sum of the first n natural numbers.

1. 

2. 

3. 

4. 

Q.

What is the sum of $1,2,3,\cdots ,n$?

Q.

Find the sum of integers from 1 to 100 that are divisible by 2 or 5.

Q.

How many three digit number are divisible by $7$?

Q.

Find the sum of first $10$ natural numbers.

Q.

Find the ${31}^{\mathrm{st}}$ term of the AP whose ${11}^{\mathrm{th}}$ term is $38$ and the ${16}^{\mathrm{th}}$ term is $73$.

Q. The sum of all natural numbers ${}^{\prime }{n}^{\prime }$ such that $100<n<200$ and H.C.F. $(91,n)>1$ is:

1. $3203$
2. $3221$
3. $3121$
4. $3303$

Q.

Determine the A.P. whose third term is $16$ and the $7th$ term exceeds the $5th$ term by $12$.

Q.

Let $m$ and $M$ be respectively the minimum and maximum values of

$\left|\begin{array}{l}\begin{array}{ccc}{\cos }^{2}x& 1+{\sin }^{2}x& \sin 2x\\ 1+{\cos }^{2}x& {\sin }^{2}x& \sin 2x\\ {\cos }^{2}x& {\sin }^{2}x& 1+\sin 2x\end{array}\end{array}\right|$

Then the ordered pair $(m,M)$ is equal to:

1. $(-3,-1)$

2. $(-4,-1)$

3. $(1,3)$

4. $(-3,3)$

Q.

Let $A=\left\{X={\left(x,y,z\right)}^T:PX=0\text{and}{x}^2+y^2+z^2=1\right\}$ where $P=\left[\begin{array}{ccc}1& 2& 1\\ -2& 3& -4\\ 1& 9& -1\end{array}\right]$ then the set $A$:

1. contains more than two elements.

2. is a singleton.

3. contains exactly two elements.

4. is an empty set

Q.

If the ${10}^{th}$ term of a geometric progression is 9 and $4^{th}$ term is 4, then its $7^{th}$ term is

1. $6$

2. $36$

3. $\frac{4}{9}$

4. $\frac{9}{4}$

Q.

How many multiples of $4$ lie between $10$ and $250$$?$

Q.

$\frac{5}{1·2·3}+\frac{7}{3·4·5}+\frac{9}{5·6·7}+...$ is equal to

1. $\log \left(\frac{8}{e}\right)$

2. $\log \left(8e\right)$

3. $\log \left(\frac{e}{8}\right)$

4. none of these

Q. Let $m$ arithmetic means are inserted between $1$ and $31$. If the ratio of the $7^{\text{th}}$ mean and $(m-1{)}^{\text{th}}$ mean is $5:9$, then which of the following is/are correct?

1. The value of $m$ is $14$.
2. The value of $m$ is $16$.
3. The sum of all mean's is $256$.
4. The sum of all mean's is $224$.

Q. The sum of n terms of two A.P. are in the ratio 5n+4:9n+6.
The ratio of their 18th terms is

1. 47:84
2. 179:321
3. 89:159
4. 2:3

Q.

The quadratic equations $x^2-6x+a=0,x^2-cx+6=0$ have one root in common. The other roots of the first and second equations are integers in the ratio $4:3$. Then, the common root is

1. $2$

2. $1$

3. $4$

4. $3$

Q. The sum of all two digit positive numbers which when divided by $7$ yield $2$ or $5$ as remainder is :

1. $1465$
2. $1356$
3. $1256$
4. $1365$

Q.

The smallest positive integer $n$ for which ${\left(\frac{1+i}{1-i}\right)}^n=1$ is,

1. $n=8$

2. $n=16$

3. $n=12$

4. None of these

Q.

If the $9^{th}$ term of an A.P be zero, then the ratio of its ${29}^{th}$ and ${19}^{th}$ term is

1. $1:2$

2. $2:1$

3. $1:3$

4. $3:1$

Q. The angle between the asymptotes of the hyperbola $x^2–3y^2=3$ is

1. 
2. 
3. 
4. 

Q.

The $12th$ term of an AP is $-13$ and the sum of its first four terms is $24$. Find the sum of its first $10$ terms.

Q.

The four arithmetic means between $3$ and $23$ are

1. $5,9,11,13$

2. $7,11,15,19$

3. $5,11,15,22$

4. $7,15,19,21$

Q. One side of an equilateral triangle is $24$ cm. The midpoints of its sides are joined to form another triangle whose midpoints are in turn joined to form still another triangle this process continues indefinitely. The sum of the perimeters of all the triangles is

1. $144\text{cm}$
2. $130\text{cm}$
3. $142\text{cm}$
4. $140\text{cm}$

Q.

If $a+b+c=0$, then the roots of the equation$4a{x}^2+3bx+2c=0$ are

1. $Equal$

2. $Imaginary$

3. $Real$

4. $Noneofthese$

Q. Let there are odd number of stones placed one by one at intervals of $10\text{m}$ along a road. These stones have to be assembled around the middle stone. A person can carry only one stone at a time. A man carried out the job with one of the end stones by carrying them in succession. In carrying all the stones, he covered a distance of $3\text{km}$. Then the number of stones is

1. $12$
2. $25$
3. $21$
4. $31$

Q.

How many terms of the sequence $\sqrt{3},3,3\sqrt{3},....$ must be taken to make the sum $39+13\sqrt{3}$ ?

Q. If fourth term of a G.P. is 3, the product of the first seven terms is

1. 
2. 
3. 
4. 

Q.

$1,5,14,30,55,91,?$ $,$ Find the missing term$.$

1. $130$

2. $140$

3. $150$

4. $160$

Q. The sum of the mean and variance of a binomial distribution is $15$ and the sum of their squares is $117$. Probability of atmost one success in case of these trials will be:

1. $\frac{31}{3}{\left(\frac{1}{3}\right)}^{26}$
2. $5{\left(\frac{2}{3}\right)}^{26}$
3. $\frac{10}{3}{\left(\frac{2}{3}\right)}^{26}$
4. $\frac{29}{3}{\left(\frac{2}{3}\right)}^{26}$