Arithmetic Mean

Q. If two consecutive numbers are removed from $1,2,3,\dots ,50$, the arithmetic mean becomes $\frac{105}{4}$, then the removed number are

1. $7$ and $8$
2. $8$ and $9$
3. $9$ and $10$
4. $10$ and $11$

Q. The arithmetic mean of six numbers is $\mathrm{20.5.}$ If two numbers are added to the progression, then the new arithmetic mean is $\mathrm{17.5.}$ If their difference is $3,$ then the two numbers are

1. $7$ and $10$
2. $-7$ and $10$
3. $8$ and $11$
4. $6$ and $10$

Q. The absolute difference of the two positive numbers whose arithmetic mean is $34$ and the geometric mean is $16,$ is

1. $64$
2. $56$
3. $60$
4. $68$

Q. If $A_1,A_2$ be two arithmetic means between $\frac{1}{3}$ and $\frac{1}{24},$ then their values are

1. $\frac{7}{36},\frac{5}{72}$
2. $\frac{5}{72},\frac{17}{72}$
3. $\frac{17}{72},\frac{5}{36}$
4. $\frac{7}{72},\frac{5}{36}$

Q.

Five numbers are in A.P., whose sum is $25$ and product is $2520$.

1. $16$

2. $27$

3. $7$

4. $\frac{21}{2}$

Q. Let $a_1,a_2,a_3,\dots$ be terms of an A.P. If $\frac{a_1+a_2+\cdots +a_p}{a_1+a_2+\cdots +a_q}=\frac{p^2}{q^2}(p\ne q)$, then $\frac{a_6}{a_{21}}$ is equal to

1. $\frac{41}{11}$
2. $\frac{11}{41}$
3. $\frac{31}{11}$
4. $\frac{11}{31}$

Q. If A be an arithmetic mean between two numbers and S be the sum of n arithmetic means between the same numbers, then

1. S - nA
2. A = nS
3. A = S
4. None of these

Q. If $A_1,A_2$ be two arithmetic means between $\frac{1}{3}$ and $\frac{1}{24},$ then their values are

1. $\frac{7}{72},\frac{5}{36}$
2. $\frac{17}{72},\frac{5}{36}$
3. $\frac{7}{36},\frac{5}{72}$
4. $\frac{5}{72},\frac{17}{72}$

Q. If $n$ arithmetic mean's are inserted between $20$ and $80$, such that the ratio of first mean to the last mean is $1:3$, then the value of $n$ is

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. Let $\alpha ,\beta \in R$. If $\alpha ,{\beta }^2$ be the roots of quadratic equation $x^2-px+1=0$ and ${\alpha }^2,\beta$ be the roots of quadratic equation $x^2-qx+8=0$, then the value of ${}^{\prime }{r}^{\prime }$ if $\frac{r}{8}$ be arithmetic mean of $p$ and $q$, is

1. $\frac{83}{2}$
2. $83$
3. $\frac{83}{8}$
4. $\frac{83}{4}$

Q. A number is the reciprocal of the other. If the arithmetic mean of the two numbers be $\frac{13}{12},$ then the numbers are

1. $\frac{1}{4},\frac{4}{1}$
2. $\frac{3}{4},\frac{4}{3}$
3. $\frac{2}{5},\frac{5}{2}$
4. $\frac{3}{2},\frac{2}{3}$

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. $353$
3. $391$
4. $412$

Q. Let a, b, c be positive integers such that $\frac{b}{a}$ is an integer. If a, b, c are in geometric progression and the arithmetic mean of a, b, c is b + 2, then the value of $\frac{a^2+a-14}{a+1}$ is___

Q. The arithmetic mean of six numbers is $\mathrm{20.5.}$ If two numbers are added to the progression, then the new arithmetic mean is $\mathrm{17.5.}$ If their difference is $3,$ then the two numbers are

1. $-7$ and $10$
2. $6$ and $10$
3. $8$ and $11$
4. $7$ and $10$