Question-69
a) Find a formula for 
By examining the values of this expression for small values of n.
The formula for the sum is ?
b) Prove the formula you conjectured in part (a).
Solution
a) 
b) We prove this by induction. It is clear that 
for n = 1. We want to show that
![[\frac{5}{1.2} + \frac{5}{2.3} + .... + \frac{5}{k(k + 1)}]+\frac{5}{(k+1)(k+2)}= \frac{5(k+1)}{k+2}](https://mymathangels.com/wp-content/ql-cache/quicklatex.com-08c77a95b9915440f256ffadd7bf8fd0_l3.png "Rendered by QuickLaTeX.com")
, Starting from the left , we replace the quantity in brackets 
(by the inductive hypothesis), and then do the algebra 
, yielding the desired expression.