If $a_1,a_2,a_3.......a_{2n+1}$ are in AP, then

Question

Value of $$\frac{a_{2n+1}-a_1}{a_{2n+1}+a_1} + \frac{a_{2n}-a_2}{a_{2n}+a_2} + .... \frac{a_{n+2}-a_n}{a_{n+2}+a_n}$$

WHAT I DID

The denominators will all be equal. So it’s simply addition of $$\frac{a_{2n+1}+a_{2n}+.....a_{n+2}-(a_1+a_2+a_3.....a_n)}{a_{2n+1}+a_1}$$ Divided by a common denominator. How should I solve further?

Answer 1

Hint:

For $2\le r\ln n,$

$$\dfrac{a_{2n+2-r}-a_r}{a_{2n+2-r}+a_r}=d\cdot\dfrac{(2n+2-r-1)-(r-1)}{2a_1+d(2n+2-r-1+r+1)}=?$$ where $d$ is the common difference

So, the denominator $$=2a_{n+2}$$