Question

Consider the series x_(n+1) = x_n/2 + 9/(8x_n), with x₀ = 0.5, obtained from the Newton-Raphson method. The series converges to:

Accepted Answer

The correct answer is (A) 1.5. Newton-Raphson for finding √c via f(x)=x²−c gives the iteration x_(n+1) = x_n/2 + c/(2x_n). Matching the given recurrence's second term 9/(8x_n) = c/(2x_n) gives c/2 = 9/8, so c = 9/4 = 2.25, and the series converges to √2.25 = 1.5. (Direct iteration confirms it: x₀=0.5 → x₁=2.5 → x₂=1.7 → x₃≈1.512 → x₄≈1.500, converging to 1.5.)

Explanation