Question

Consider the first order difference equation \[ x_n = \left( \frac{n + 1}{n} \right) x_{n - 1}, \quad n = 1, 2, 3, \ldots \] If $x_0 = 2$, then $x_{100} - x_{50}$ equals __________. (in integer)

Hint:

For multiplicative difference equations, write out a few terms to recognize the telescoping or factorial-like pattern.

Answer 1

Correct Answer: 100

Step 1: Expand recurrence relation.
\[ x_1 = \frac{2}{1}x_0, \quad x_2 = \frac{3}{2}x_1, \quad x_3 = \frac{4}{3}x_2, \ldots \] \[ x_n = \frac{n+1}{n} \times \frac{n}{n-1} \times \ldots \times \frac{2}{1} \, x_0 = (n+1)x_0. \]
Step 2: Compute required terms.
\[ x_{100} = 101x_0 = 202, \quad x_{50} = 51x_0 = 102. \]
Step 3: Find the difference.
\[ x_{100} - x_{50} = 202 - 102 = 100. \]
Step 4: Conclusion.
\[ \boxed{x_{100} - x_{50} = 100} \] (in integer).