R D Sharma Solutions for Chapter: Arithmetic Progressions, Exercise 6: EXERCISE 19.6

Find the A.M. between: $7$ and $13$

Find the A.M. between $12$ and $-8$

Find the A.M. between $(x-y)$ and $(x+y)$.

There are $\mathrm{n}$ A.M.s between $3$ and $17$. The ratio of the last mean to the first mean is $3:1$. Find the value of $\mathrm{n}$.

Insert  A.M.s between $7$ and $71$in such a way that the $5^{\mathrm{th}}$ A.M. is $27$. Find the number of A.M.s.

If $\mathrm{n}$ A.M.s are inserted between two numbers, prove that the sum of the means equidistant from the beginning and the end is constant.

If $\mathrm{x}, \mathrm{y}, \mathrm{z}$ are in A.P. and ${\mathrm{A}}_1$ is the A.M. of $\mathrm{x}$ and $\mathrm{y}$ and ${\mathrm{A}}_2$ is the A.M. of $\mathrm{y}$ and $\mathrm{z}$, then prove that the A.M. of ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$ is $\mathrm{y}$.

Insert five numbers between $8$ and $26$ such that the resulting sequence is an $\mathrm{A}.\mathrm{P}.$.