If the positive integers are written in a triangular array as shown below,
then the row in which the number `2010` will be, is

Options

A. `65`
B. `61`
C. `63`
D. `65`

Answer

C

Explanation

`(c )` Let `2010` be in `k^(th)` row
`impliesk^(th)` term of series `1,2,4,7,…… le 2010` (series formed by `1^(st)` term of each group)
and `(k+1)^(th)` term of series `1,2,4,7,… gt 2010`
`S_(n)=1+2+4+7+...+T_(n)`
`S_(n)=1+2+4+...+T_(n-1)+T_(n)`
`thereforeoverline(0=1+(1+2+3+...(n-1)"terms"-T_(n))`
`implies T_(n)=(n^(2)-n+2)/(2)`
`implies (k^(2)-k+2)/(2) le 2010` and `(k^(2)+k+2)/(2) gt 2010`
`impliesk^(2)-k-4018 le 0` and `k^(2)+j-4018 gt 0`
`implies(k-(1)/(2))^(2) le (16073)/(4))` and `(k+(1)/(2))^(2) gt (16073)/(4)`
`k-(1)/(2) le 63.3` and `k+(1)/(2) gt 63.3`
`impliesk=63`