In an $\mathrm{A}.\mathrm{P}.$ the third term is four times the first term and the sixth term is $17$, find the series.

Find the number of integers between $100 \&1000$ that are:

(i) divisible by $7$               

(ii) not divisible by $7$

If $a, b, c$ are in A.P., then show that $a^2\left(b+c\right), b^2\left(c+a\right), c^2\left(a+b\right)$ are also in A.P.

If $a, b, c$ are in A.P., then show that:

$b+c-a, c+a-b, a+b-c$ are in A.P.