Arithmetic Mean

Q. If $A=\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]$ and $M=A+A^2+A^3+.....+A^{20}$, then the sum of all the elements of the matrix $M$ is equal to

Q.

Let $a_{1,}{a}_2,a_3,..$ be terms of an A.P. whose common difference is an integer and $S_n=a_1+a_2+...+a_n$. If $a_1=1,a_n=300$ and $15\le n\le 50,$then the ordered pair $\left(S_{n-4},a_{n-4}\right)$ is equal to:

1. $\left(2480,248\right)$

2. $\left(2480,249\right)$

3. $\left(2490,249\right)$

4. $\left(2490,248\right)$

Q.

The sum of the first three terms of a G.P. is $S$ and their product is $27$. Then all such $S$ lie in

1. $(-\infty ,-9]\cup [3,\infty )$

2. $[-3,\infty )$

3. $(-\infty ,-9]$

4. $(-\infty ,-3]\cup [9,\infty )$

Q.

Which of the following statements is always true?

1. $\frac{(x-y)}{2}$is a rational number between $x$ and$y$.

2. $\frac{(x+y)}{2}$ is a rational number between $x$ and$y$.

3. $\frac{(x\times y)}{2}$is a rational number between $x$ and$y$.

4. $\frac{(x÷y)}{2}$is a rational number between $x$ and$y$.

Q.

How do you add all the odd numbers between $1-99$ inclusive?

Q.

If $f\left(x+y,x-y\right)=xy$ , then the arithmetic mean of $f\left(x,y\right)$ and $f\left(y,x\right)$ is

1. $x$

2. $y$

3. $0$

4. $1$

Q.

Find the two numbers whose A.M. is 25 and GM is 20.

Q.

The sum of numbers between 100 and 500, which are divisible by 6, is:

1. 67

2. 498

3. 2010

4. 20100

Q.

The sum of all odd numbers between $1$ and $1000$ which are divisible by $3$, is

1. $83667$

2. $90000$

3. $83660$

4. None of these

Q.

The sum of $n$ arithmetic means between$a$ and $b$ is

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

2. $n(a+b)$

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

4. $(n+1)(a+b)$

Q.

If a, b, c are in A.P. and x. y, z are in G.P., then the value of $x^{b-c}{y}^{c-a}{z}^{a-b}$ is

1. $x^a$ $y^b$ $z^c$

2. 0

3. 1

4. xyz

Q. Find the ${10}^{th}$ and $n^{th}$ terms of the G.P. $5,25,125,...$

Q.

If n arithmetic means are inserted between 1 and 31 such that the ratio of the first mean and nth mean is 3 : 29, then the value of n is

1. 10

2. 14

3. 13

4. 12

Q.

If the sum of the second, third and fourth terms of a positive term G.P. is $3$ and the sum of its sixth, seventh and eighth terms is $243$, then the sum of the first $50$ terms of this G.P. is:

1. $\frac{2}{13}\left(3^{50}-1\right)$

2. $\frac{1}{26}\left(3^{49}-1\right)$

3. $\frac{1}{13}\left(3^{50}-1\right)$

4. $\frac{1}{26}\left(3^{50}-1\right)$

Q. If an even number of A.M.s are inserted between two numbers whose sum is $\frac{13}{6}$, such that their sum exceeds their numbers by unity, then the number of means is

Q.

If the equation $2a{x}^2-3bx+4c=0$ and $3x^2-4x+5=0$have a common root, then $\frac{(a+b)}{(b+c)}$ is equal to $(a,b,c\in \mathrm{ℝ})$

1. $\frac{1}{2}$

2. $\frac{3}{35}$

3. $\frac{34}{31}$

4. $\frac{29}{23}$

5. None of these

Q.

Find the mean of first five multiples of $3$.

Q.

If x, y, z are in A.P. and $A_1$ is the A.M. of x and y and $A_2$ is the A.M. of y and z, then prove that the A.M. of $A_1$ and $A_2$ is y.

Q. If $a+b+c=1$, where $a,b,c$ are positive real numbers, then the minimum value of $\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}$ is

1. $13$
2. $24$
3. $27$
4. $32$

Q.

Find the sum of even numbers between$1$ and$25$.

Q.

If the arithmetic mean of $7,9,11,13,x,21$ is $13$, find the value of $x$.

Q. If n arithmetic means are inserted between two sets of numbers a, 2b and 2a, b, where $a,bϵR.$ Suppose that $m^{th}$ arithmetic mean between these two sets of numbers is same, then the ratio a:b equals

1. n - m + 1:m
2. n - m + 1:n
3. m:n - m + 1
4. n:n - m + 1

Q. The equation $P\left(x\right)=x^3+a{x}^2+bx+c=0$ has the property that the arithmetic mean of its roots, the product of its roots and the sum of its coefficients are all equal. If the $y$-intercept of the graph of $y=P\left(x\right)$ is $2,$ then the value of $b+P\left(2\right)$ is

Q.

Complete the following series: $0,5,22,57,?,205.$

Q. How many terms of the $A.P.-6,-\frac{11}{2},-5,\dots$ are needed to give the sum $-25$ ?

Q. If $A=\{x|x\in N,x\le 5\},$$B=\{x|x\in Z,x^2-5x+6=0\},$ then the number of onto functions from $A$ to $B$ is

1. 30
2. 23
3. 32
4. 2

Q. Let $f$ be defined for all non-zero real numbers $x$ as $f\left(x\right)+2f\left(\frac{1}{x}\right)=3x.$ Then the number of values of $x$ satisfying the equation $f\left(x\right)=f(-x)$ is

Q. Let $a_1,a_2,a_3,\dots$ be an A.P. such that $a_6+a_9+a_{12}+a_{15}=20$. The value of sum of the first $20$ terms is equal to

1. $100$
2. $140$
3. $150$
4. $200$

Q.

What is the sum of the first $10$ multiples of $6$?

Q. If the sum of $n$ terms of an A.P. is given by $S_n=3n^2-4n,$ then its ${50}^{th}$ term is

1. $293$
2. $391$
3. $412$
4. $353$