Question:hard

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

Updated On: Jan 15, 2026
  • -5151
  • -5150
  • -5051
  • -5050
Show Solution

The Correct Option is D

Solution and Explanation

Provided Information:

A recursive relation: \[ x_{m+1} = x_m - (m + 1) \] with initial condition: \[ x_1 = -1 \]

Calculation Procedure:

Applying the recurrence relation:

  • \[ x_2 = x_1 - 2 = -1 - 2 = -3 \]
  • \[ x_3 = x_2 - 3 = -3 - 3 = -6 \]
  • \[ x_4 = x_3 - 4 = -6 - 4 = -10 \]
  • The pattern continues...

Observation of the pattern yields: \[ x_n = - (1 + 2 + 3 + \dots + n) = -\frac{n(n+1)}{2} \]

Final Determination:

Calculating: \[ x_{100} = -\frac{100 \times 101}{2} = -5050 \]

Result:

\(\boxed{-5050}\)

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