Question

Value of 1 + 1/(1+2) +1/ (1+2+3)..upto 1/ (1+2+3…n) is?

Answer 1

TO FIND:

1/1 + 1/3 + 1/6 + 1/10 + …………. 1/k =?

Here each term can be written with a pattern as follows:

1st term : (2/1) - (2/2) = 1/1

2nd term : (2/2) - ( 2/3) = 1/3

3rd term: ( 2/3) - (2/4) = 1/6

4th term = ( 2/4) - ( 2/5) = 1/10

……………….

nth term = (2/n) - (2 / n+1 )

Now, let's find the sum of the 4 terms :

2/1 - 2/2 + 2/2 - 2/3 + 2/3 - 2/4 + 2/4 - 2/5

we get, 2/1 - 2/5 ( as all other terms are cancelled out)

So the sum of all n terms =

We get, (2/1) - 2 / (n+1)

= 2 - 2/ n+1

= { 2(n+1) - 2 } / (n+1)

= {2n +2 -2 } / ( n+1)

= 2n / (n+1) ……………ANS