JIPMAT 2022 (QA) - The sum of n- terms of sequence 1/1 x 2+1/2 x 3+1/3 x 4 . Is | PYQs + Solutions

JIPMAT 2022

Algebra

Progression & Series

Easy

The sum of $n-$ terms of sequence $\frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4} \ldots \ldots$ . Is

\frac{1}{n+1}

\frac{1}{n}

\frac{n+1}{n}

\frac{n}{n+1}

Correct Option: 4

Looking at first few terms of sequence:

\frac{1}{1×2} + \frac{1}{2×3} + \frac{1}{3×4} + ... + \frac{1}{n×(n+1)}

Each term can be split using partial fractions:

\newline

\frac{1}{k×(k+1)} = \frac{1}{k} - \frac{1}{k+1}

Sum =

(\frac{1}{1} - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{4}) + ... + (\frac{1}{n} - \frac{1}{n+1})

Cancelling terms:

\frac{1}{1} - \frac{1}{n+1} = \frac{n}{n+1}

Therefore, sum of n terms =

\frac{n}{n+1}