J P Mohindru and Bharat Mohindru Solutions for Chapter: Arithmetic Progressions, Exercise 5: EXERCISE

Insert $3$ arithmetic means between $3$ and $15$.

Insert $3$ arithmetic means between $5$ and $21$.

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

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

If $A_1$ and $A_2$ are two arithmetic means between $a$ and $b$, then prove that $(2A_1-A_2)(2A_2-A_1)=ab$.

If $A_1$ and $A_2$ are two A.M.'s between $a$ and $b$, then prove that $A_1+A_2=a+b$.

Find $n$, if $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ may be the A.M. between $a$ and $b$.

Find $\text{'}n\text{'}$ so that: $\frac{a^n+b^n}{a^{n-1}+b^{n-1}}$ may be the A.M. between $a$ and $b$.