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

Given:

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

Step-by-step Calculation:

Using the recurrence:

• \[ x_2 = x_1 - 2 = -1 - 2 = -3 \]
• \[ x_3 = x_2 - 3 = -3 - 3 = -6 \]
• \[ x_4 = x_3 - 4 = -6 - 4 = -10 \]
• and so on...

From the pattern, we see: \[ x_n = - (1 + 2 + 3 + \dots + n) = -\frac{n(n+1)}{2} \]

Final Calculation:

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

Correct Answer:

\(\boxed{-5050}\)

Answer 2

Given:

\[ x_1 = -1, \quad \text{and} \quad x_m = x_{m+1} + (m+1) \] Rearranging the recursive formula: \[ x_{m+1} = x_m - (m+1) \]

Step-by-Step Calculation:

Let’s compute the first few values:

• \( x_1 = -1 \) (Given)
• \( x_2 = x_1 - 2 = -1 - 2 = -3 \)
• \( x_3 = x_2 - 3 = -3 - 3 = -6 \)
• \( x_4 = x_3 - 4 = -6 - 4 = -10 \)
• \( x_5 = x_4 - 5 = -10 - 5 = -15 \)
• \( x_6 = x_5 - 6 = -15 - 6 = -21 \)
• ... and so on ...

Clearly, the pattern becomes: \[ x_n = - (1 + 2 + 3 + \dots + n) \] Which is the negative of the sum of the first \( n \) natural numbers.

General Formula:

\[ x_n = -\frac{n(n+1)}{2} \]

Find \( x_{100} \):

\[ x_{100} = -\frac{100(101)}{2} = -50 \times 101 = \boxed{-5050} \]

Final Answer:

The correct option is (D): \[ \boxed{-5050} \]