Consider the sequence xₙ₊₁ = xₙ/2 + 9/(8xₙ), with x₀ = 0.5, obtained from the…
2007
Consider the sequence xₙ₊₁ = xₙ/2 + 9/(8xₙ), with x₀ = 0.5, obtained from the Newton-Raphson method. The sequence converges to:
- A.
1.5
- B.
√2
- C.
1.6
- D.
1.4
Show answer & explanation
Correct answer: A
Let the sequence converge to L. Since x₀ = 0.5 and the recurrence keeps the terms positive, the limiting value must be positive.
At convergence, xₙ₊₁ and xₙ both approach L. Therefore,
L = L/2 + 9/(8L).
Subtract L/2 from both sides:
L/2 = 9/(8L).
Multiplying by 8L gives:
4L² = 9
L² = 9/4
L = ±3/2.
Because the terms are positive, the limit is L = 3/2 = 1.5.