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 $b$.

Find three numbers in A.P whose sum is $21$ and product is $315$.

Find three numbers in A.P. whose sum is $24$ and product is $440$.

The sum of the first three terms of an A.P. is $36$ while their product is $1620$. Find the A.P.

Split $69$ into three parts in A.P. such that the product of the two smaller parts is $483$.

The sum of four numbers in A.P. is $4$ and their product is $385$. Find the numbers.