in sequence 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5......... find 150th term

Hope this will help you ,

this series can be converted into an easier one if we write the frequencies of the given nos. 

let'' start with a smaller no. , say the 5^th term. 

thus, the cumulative frequency of n^th no. will be the sum of the n terms,i.e., {n(n+1)}/2.

so, coming to our example, equating 7 to the sum given:-

5= {n(n+1)}/2  or n² +n -10=0

this gives:- n= [-1+(1+40)^1/2]/2     {since n cannot be -ve, so +ve sqrt.}

this gives, 3>n>2.   thus,  the cumulative frequency of 2 is less than 5 and that of 3 is more than 5.  so, n=3.  which is correct 

now, coming back to the problem , we equate 150 to the sum given  

thus, 150= {n(n+1)}/2 or n² +n - 300=0 

Now factorise and conside the +ve root and not the -ve root .

Didn''t it help you?