M L Aggarwal Solutions for Chapter: Sequences and Series, Exercise 4: EXERCISE 10.4

Find the arithmetic mean between $-3$ and $1$.

Insert five numbers between $8$ and $6$ such that the resulting sequence is an $A.P.$

Insert $6$ arithmetic means between $3$ and $24$.

Insert $10$ $A.M\text{'}s$ between $-5$ and $17$ and prove that their sum is $10$ times the $A.M.$ between $-5$ and $17$.

If $A$ is the $\mathrm{AM}$ between $a$ and $b$, prove that $(\mathrm{A}-a{)}^2+(\mathrm{A}-b{)}^2=\frac{1}{2}(a-b{)}^2$.

If $A$ is the $A.M.$ between $a$ and $b$, prove that $4(a-\mathrm{A})(\mathrm{A}-b)=(a-b{)}^2$.

Arithmetic means are inserted between $7$ and $71$ in such a way that the $5^{th}$ $A.M.$ is $27$. Find the number of $A.M\text{'}s$ inserted.

There are $n$ arithmetic means between $3$ and $17$. The ratio of the first mean to the last mean is $1: 3$. Find $n$.