Question

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

• A. -5151
• B. -5150
• C. -5051
• D. -5050

Answer 1

The Correct Option is D

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:

\(\boxed{-5050}\)