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gopi
1,2,2,3,3,3,4,4,4,4,------- what is the 100th term
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Raj
The series is  1,  (2,2), (3,3,3), (4,4,4,4), ...

1st term : 1
2nd, 3rd terms : 2
4th, 5th, 6th terms : 3
7th, 8th, 9th, 10th terms :  4
11th, 12th, 13th, 14th, 15th terms : 5

Each group ends in a sequence 1,3,6,10,15, ...
This can be represented using $\dfrac{n(n+1)}{2}$ where n=1,2,3,...
With this, we can easily find out the answer.

Say,n=7.
$\dfrac{7×8}{2}=28$
This means, 28th term is 7
=> 22nd to 28th terms will be 7

Therefore, we find largest integer value of n satisfying
$\dfrac{n(n+1)}{2} \lt 100\\ n(n+1) \lt 200$

Largest integer value of n satisfying the above condition is 13
(because 13×14=182 < 200 and if n=14, 14×15 will become greater than 200)

For n=13, $\dfrac{n(n+1)}{2}=\dfrac{13×14}{2}=91$
i.e., 91th term is 13

92nd term is 14
=> 92nd to 105 terms are 14 (taken 14 terms from 92, including 92)

Therefore 100th term is 14
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