Arithmetic Progression

Q.

Let $R_1$ and $R_2$ be two relations defined as follows:

$R_1=\left\{\left(a,b\right)\in R^2:a^2+b^2\in \mathrm{\mathbb{Q}}\right\}$and

$R_2=\left\{\left(a,b\right)\in R^2:a^2+b^2\notin \mathrm{\mathbb{Q}}\right\}$, where $\mathrm{\mathbb{Q}}$ is the set of all rational numbers. Then :

1. $R_1$ is transitive but $R_2$ is not transitive

2. $R_1$ and $R_2$ are both transitive

3. $R_2$ is transitive but $R_1$ is not transitive

4. Neither $R_1$ nor $R_2$ is transitive

Q.

Find the sum of the first $100$ natural numbers.

Q.

The sum of $1+3+5+7+..........$ upto $n$ terms is

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

2. ${\left(2n\right)}^2$

3. $n^2$

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

Q. Let $p,q$ be the roots of the equation $m{x}^2+x(2-m)+3=0$. Let $m_1,m_2$ be the two values of $m$ satisfying the equation $\frac{p}{q}+\frac{q}{p}=\frac{2}{3}$. The value of $\frac{m_1}{m_2^2}+\frac{m_2}{m_1^2}$ is

1. $99$
2. $-99$
3. $96$
4. $0$

Q.

If $1,{\log }_{10}\left(4^x-2\right)$ and ${\log }_{10}\left[4^x+\left(\frac{18}{5}\right)\right]$ are in arithmetic progression for a real number $x$, then the value of the determinant

$\left|\begin{array}{ccc}2\left[x-\frac{1}{2}\right]& x-1& x^2\\ 1& 0& x\\ x& 1& 0\end{array}\right|$ is equal to

Q.

If $S_n$ denotes the sum of $n$ terms of an arithmetic progression, then the value of $\left(S_{2n}-S_n\right)$ is equal to

1. $2S_n$

2. $S_{3n}$

3. $\frac{S_{3n}}{3}$

4. $\frac{S_n}{2}$

Q.

If $a^2,b^2,c^2$ are in A.P. prove that $\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}$ are in A.P.

Q. The sum of $n$ terms of two arithmetic progressions are in the ratio $(3n+8):(7n+15)$. Find the ratio of their ${12}^{th}$ terms.

Q.

The angles of a quadrilateral are in A.P. and the greatest angle is

${120}^{\circ }$. Express the angles in radians.

Q. If $a_1,a_2,a_3,\cdots ,a_n$ are in A.P. and $a_1+a_4+a_7+\cdots +a_{16}=114$ , then $a_1+a_6+a_{11}+a_{16}$ is equal to :

1. $76$
2. $98$
3. $38$
4. $64$

Q.

If a, b, c are in A.P., then show that:
(i) $a^2(b+c),b^2(c+a),c^2(a+b)$ are also in A.P.
(ii) $b+c-a,c+a-b,a+b-c$ are in A.P.
(iii) $bc-a^2,ca-b^2,ab-c^2$ are in A.P.

Q.

Find the 12th term from the end of the following arithmetic progressions:

(i) 3, 5, 7, 9, ...... 201

(ii) 3, 8, 13, ....... 253

(iii) 1, 4, 7, 10, ....... 88

Q.

(i) How many terms are there in the A.P. 7, 10, 13, ..... 43 ?

(ii) How many terms are there in the A.P. $-1,\frac{-5}{6}-\frac{2}{3},\frac{1}{2},\dots \dots \frac{10}{3}$ ?

Q.

If $lo{g}^2,log(2^x-1)$ and $log(2^x+3)$ are in A.P., write the value of x.

Q.

What is the total sum of $1$ to $30$?

Q.

The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21, from these numbers in that order, we obtain an A.P. Find the numbers.

Q. Let $p,q\in R$. If $2-\sqrt{3}$ is a root of the quadratic equation $x^2+px+q=0$, then

1. $p^2-4q+12=0$
2. $q^2-4p-16=0$
3. $p^2-4q-12=0$
4. $q^2+4p+14=0$

Q.

If $a(\frac{1}{b}+\frac{1}{c}),b(\frac{1}{c}+\frac{1}{a}),c(\frac{1}{a}+\frac{1}{b})$ are in A.P., prove that a, b, c are in A.P.

Q. If $2\log (x+1)-\log (x^2-1)=\log 2,$ then $x=$

1. only $3$
2. $-1$ and $3$
3. only $-1$
4. $1$ and $3$

Q.

If the angles of a triangle are in A.P., then the measures of one of the angles in radians is

1. $\frac{\pi }{6}$

2. $\frac{\pi }{3}$

3. $\frac{\pi }{2}$

4. $\frac{2\pi }{3}$

Q.

Given $d=5$ and $S_9=75$. Find the value of $a$ and $a_9$

Q.

The angles of a triangle are in A.P. such that the greatest is 5 times the least. Find the angles in radians.

Q.

An AP consists of $23$terms. If the sum of the three terms in the middle is $141.$ and the sum of the last three terms is$261$, then the first term is

1. $6$

2. $5$

3. $4$

4. $3$

Q. Let $(1+x^2{)}^2(1+x{)}^n=\sum _{k=0}^{n+4}{a}_k{x}^k$. If $a_1,a_2,a_3$ are in arithmetic progression, then the sum of all the possible values of n is

Q.

For $a,b\in R$ defined by $a*b=\frac{a}{(a+b)}$, where $a+b\ne 0$. If $a*b=5$, then the value of $b*a$ is

1. $5$

2. $-4$

3. $-5$

4. $4$

Q.

If the first, second and last terms of an A. P. are$a,b$ and $2a$ respectively, the sum of the series is

1. $\frac{ab}{2(b-a)}$

2. $\frac{3ab}{2(b-a)}$

3. $\frac{ab}{(b-a)}$

4. None of these

Q.

Find $7+10\frac{1}{2}+14+...+84$

Q.

Find the next term of the AP: $3,1,-1,-3,...$

Q.

The least positive integer $n$, which will reduce ${\left[\frac{\left(1+i\right)}{\left(1-i\right)}\right]}^n$to a real number is

1. $2$

2. $3$

3. $4$

4. $5$

Q. Sum of the series 3 + 5 + 9 + 17 + 33 + ..... to n terms is

1. 
2. 
3. 
4.