Given that,
- $x_{1} = \;– 1 $
- $ x_{m} = x_{m+1} + (m+1); \; m>0 \quad \longrightarrow (1) $
We can re-write the equation $(1).$
$x_{m+1} = x_{m} – (m+1) $
Put $ m = 1,2,3,4,5, \dots $
Then,
- $ x_{2} = x_{1} – 2 \Rightarrow x_{2} = – 1 – 2 $
- $ x_{3} = x_{2} – 3 \Rightarrow x_{3} = – 1 – 2 – 3 $
- $ x_{4} = x_{3} – 4 \Rightarrow x_{4} = – 1 – 2 – 3 – 4$
- $ x_{5} = x_{4} – 5 \Rightarrow x_{5} = – 1 – 2 – 3 – 4 – 5 $
- $\vdots \;\; \vdots \;\; \vdots\;\; \vdots\;\; \vdots $
So, $x_{100} = \;– 1 – 2 – 3 – 4 – 5 – \;\dots\; – 100 $
$ \Rightarrow x_{100} =\; – ( 1 + 2 + 3 + 4 + 5 + \dots + 100 ) $
$ \Rightarrow x_{100} = \;– \left[ \frac{100(101)}{2} \right] $
$ \Rightarrow x_{100} = \;– 50 \times 101 $
$ \Rightarrow \boxed{ x_{100} =\; – 5050} $
$\therefore$ The value of $x_{100}$ is $ \;– 5050.$
Correct Answer$: \text{D}$
$\textsf{PS:}$ Sum of first $n$ natural numbers $ = 1 + 2 + 3 + 4 + 5 + \dots + n = \dfrac{n(n+1)}{2}. $