The numbers `1,3,6,10,15,21,28."……"` are called triangular numbers. Let `t_(n)` denotes the `n^(th)` triangular number such that `t_(n)=t_(n-1)+n,AA n ge 2`. The number of positive integers lying between `t_(100)` and `t_(101)` are

Options

A. 99
B. 100
C. 101
D. 102

Answer

B

Explanation

Given sequece `1,3,6,10,15,21,28,"…"`
where `t_(n)=t_(n-1)+n,AA n ge 2`
So, `t_(n)=[t_(n-2)+(n-1)]+n`
`vdots " "vdots " " vdots " "`
`t_(n)=t_(1)+2+3+"........"+(n-1)+n`
`t_(n)=1+2+3+"......"+n`
`t_(n)=(n(n+1))/(2)`
`t_(100)=(100xx101)/(2)=5050`
`t_(101)=(101xx102)/(2)=101xx51=5151`
Number of positive integers lying between `t_(100)` and `t_(101)`
`=5151-5050-1`
`=101-1=100`.