Question

Let \( u_0 = 0 \), and \( u_{n+1} = 2u_n + 1 \). What is the value of \( u_{10} \)? (nearest to)

• A. 1023
• B. 2047
• C. 4095
• D. 8195

Hint:

Look for exponential growth in recursions — test if it fits \( u_n = 2^n - 1 \) or similar.

Answer 1

The Correct Option is A

Given recurrence: \[ u_0 = 0
u_1 = 2(0) + 1 = 1
u_2 = 2(1) + 1 = 3
u_3 = 2(3) + 1 = 7
u_4 = 2(7) + 1 = 15
u_5 = 2(15) + 1 = 31
u_6 = 2(31) + 1 = 63
u_7 = 2(63) + 1 = 127
u_8 = 2(127) + 1 = 255
u_9 = 2(255) + 1 = 511
u_{10} = 2(511) + 1 = \boxed{1023} \] Pattern: This is a geometric-like recurrence. The closed-form is: \[ u_n = 2^n - 1 u_{10} = 2^{10} - 1 = 1024 - 1 = \boxed{1023} \]