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  4. Number of terms in the sequence 1, 3, 6, 10, 15, …, 5050 is

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Q. Number of terms in the sequence $1, 3, 6, 10, 15, …, 5050$ is

Sequences and Series

Solution:

Let $S=1+3+6+10+15+, \ldots,+t_{n}\,\,\, (1)$
then $S=1+3+6+10+, \ldots,+t_{n-1}+t_{n}\,\,\,\, (2)$
$(1)-(2) \Rightarrow 0=(1+2+3+4+\ldots$ to $n$ terms $)-t_{n}$
$\Rightarrow t_{n}=\frac{n(n+1)}{2}$
Given, $5050=\frac{n(n+1)}{2} $
$\Rightarrow n^{2}+n-10100=0$
$\Rightarrow n =\frac{-1 \pm \sqrt{1+40400}}{2} $
$=\frac{-1 \pm \sqrt{40401}}{2} $
$=\frac{-1 \pm 201}{2}=-101,100 $
$\therefore n=100 . $
$(\because n$ is a positive integer $)$