Question-69

Question-69

a) Find a formula for  \frac{5}{1.2} + \frac{5}{2.3} + ….+ \frac{5}{n(n+1)}

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) \frac{5n}{n+1}

 

b) We prove this by induction. It is clear that \frac{5n}{n+1} = \frac{5}{2} 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}, Starting from the left , we replace the quantity in brackets  \frac{5k}{k+1}   (by the inductive hypothesis), and then do the algebra \frac{5k}{k+1}+\frac{5}{(k+1)(k+2)} = \frac{5k^{2}+10k+5}{(k+1)(k+2)} = \frac{5(k+1)}{k+2} , yielding the desired expression.

 

 

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