In the sequence 1,2,2,3,3,3,4,4,4,4,...........where n consecutive terms have the value n, the 150^th term, is

a)17

b)16

c)18

d)none of these

How to solve these types of sums??

Notice that the last instance of a particular integer, i, occurs at the term that is the sum of all integers up to i.

For example, the last 3 is the sixth term and Σi (i=1 to 3)=6.  If you think about it for a bit this is obvious.

 

So let's set up an equation.  Σi (1 to m)=150.  If such an m existed, then it would be the last instance of the inetger m and the 150th term.  A little thought lures us to the inequality 150

 

150

 

300

 

We could use the quadratic formula, but I would just estimate...17 mostly from the first inequality.

 

302

 

17 is the 150th term