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

Explanation

Let the 150th term = n
Then, `1+2+3+"...."+(n-1)lt150lt1+2+3+"....."+n`
`implies ((n-1)n)/(2)lt150lt(n(n+1))/(2)`
`impliesn(n-1)lt300 lt n(n+1)`
Taking first two members
`impliesn(n-1)lt300 implies n^(2)-n- 300 lt0`
`implies (n-(1)/(2))lt300+(1)/(4)`
`implies 0 lt n lt (1)/(2)+(sqrt(1201))/(2)`
`implies 0 lt n lt 17.8 " " "......"(i)`
and taking last two members,
`n(n=1)gt300`
`implies (n+(1)/(2))^(2)gt300+(1)/(4)`
`therefore " " ngt -(1)/(2)+(sqrt1201)/(2)`
`implies " "ngt 16.8" " "....."(ii)`
From Eqs. (i)and (ii), we get
16.8ltnlt17.8
`implies " "n=17 `