If it is Arithmetic Progression then find the value of s?

The value of s = I started this question by making an A.P as the common difference is same and got the answer that I need number of terms to proceed further but my valie for number of terms is coming in fraction that is not possible i tried many times but end up with the same value so I can't proceed further as I don't know the value of n to put in the sum of Arithmetic Progression series?

Answer

Hint:

The general term is $$\frac{1}{(5r-3)(5r+2)}$$

We use partial fraction decomposition, so:

$$\frac{1}{(5r-3)(5r+2)}=\frac{A}{5r-3}+\frac{B}{5r+2}$$

Can you find $A$ and $B$? (Hint 2: $A+B=0$)

Because $A+B=0$, can you see that between the decomposition of $$\frac{1}{(5r-3)(5r+2)}$$ and $$\frac{1}{(5(r+1)-3)(5(r+1)+2)}=\frac{1}{(5r+2)(5r+7)}$$

some terms are cancelled? Figure out which ones will not be cancelled and sum them.

This method is known as telescoping