If ${x_1} = - 1$ and ${x_m} = {x_{m + 1}} + (m + 1)$ for every positive integer $m,$ then \({x

Question

If ${x_1} = - 1$ and ${x_m} = {x_{m + 1}} + (m + 1)$ for every positive integer $m,$ then ${x_{100}}$ equals

-5151
-5150
-5051
-5050

Option: 4
Solution:

${x_m} + 1 = {x_m} - (m + 1)$

${x_2} = {x_1} - 2 = - 1 - 2 = - 3$

${x_3} = {x_2} - 3 = - 1 - 2 - 3 = - 6$

Similarly,

${x_n} = - (1 + 2 + 3 + \ldots + n) = - \frac{{n(n + 1)}}{2}$

Hence ${x_{100}} = - \frac{{100(101)}}{2} = - 5050$

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