Telescoping Series 7 – Extension

We have 1/1 - 1/n+1 = 0.998

=> 1/n+1 = 0.002

=> n+1 = 500

=> n = 499

Just like I explained in the previous problem, the decimals are simply the fraction formed from the denominators. 0.998 is calculated from dividing 499 by 500. n= 499 as 500 is (499 + 1) OR (n+1)

1 - 1 /1+n = 0.998

n / n +1 =998 /1000

1000 n = 998 n + 998

2n = 998

n= 499

n/(n+1)=0.998 =>n=0.998n+0.998 =>0.002n=0.998 =>2n=998 =>n=499

There is a simple way to solve this:

First, convert 0.998 into a fraction - which is 499/500. Since the sum of the terms up to 1/n*(n+1) has to be n/n+1, the value of n=499.

$1 - \frac{1}{n+1} = \frac{499}{500}$

$\frac{1}{n+1} = \frac{1}{500}$

$n+1 = 500$

$\boxed{n = 499}$

Nice problem

separate each term, like 1/1*2=1-1/2 and so on we see that all terms except first and last cancel out and we are left with 1-1/(n+1)=0.998, now solve and get. n=499