cat 2020 Question 1

Question

If $x_{1}$ = -1 and $x_{m}$ = $x_{m + 1}$ + (m + 1) for every positive integer m, then $x_{100}$ equals

Accepted Answer

$x_1 = -1$

$x_1 = x_2 + 2 \Rightarrow x_2 = x_1 - 2 = -3$

Similarly,

$x_3 = x_1 - 5 = -6$

$x_4 = -10$

The series is -1, -3, -6, -10, -15, $\ldots$

When the differences are in AP, then the nth term is $-\frac{n(n+1)}{2}$

$x_{100} = -\frac{100(100+1)}{2} = -5050$

Question 1.