First slide

Special Series in Sequences and Series

Question

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

Moderate

A

50
B

75
C

100
D

125

Solution

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

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