CAT 2021 Slot 3 · Quant · Question 16

The recurrence gives the gap between consecutive terms: {x}_{n+1}-{x}_{n}=n-1.

Adding these gaps from n=1 up to n=99 telescopes to {x}_{100}-{x}_{1}:

{x}_{100}={x}_{1}+\sum_{n=1}^{99}{\left( n-1 \right)}=-1+\left( 0+1+2+\cdots +98 \right).

The sum is \frac{98\times 99}{2}=4851, so {x}_{100}=-1+4851=4850.

The usual slip is to add up to n=100 instead of n=99; reaching the 100th term takes only 99 steps.