Consider the function f: ℝ → ℝ defined by f(x)=(2 x)/√(1+9 x²). If the composition of f, (f ° f ° f ° … ° f)₁₀ₜᵢₘₑₛ(x)=(2¹⁰ x)/√(1+9 α x²), then the value of √(3 α+1) is equal to _______.

Solution

Maths sir · 4 steps

Do f ° f by hand, spot how the constant under the root grows, and the pattern does all 10 steps.

1. Assume the form. Suppose n copies of f give (2ⁿ x)/√(1+9cₙ x²). For one copy, c₁ = 1.
2. Apply f once more. Put g = (2ⁿ x)/√(1+9cₙ x²) into f. Then 1 + 9g² = (1+9cₙ x²+9· 4ⁿ x²)/(1+9cₙ x²). The factor √(1+9cₙ x²) cancels top and bottom.

   f(g) = (2ⁿ⁺¹ x)/√(1+9(cₙ + 4ⁿ)x²) ⇒ cₙ₊₁ = cₙ + 4ⁿ

3. Sum the pattern. So cₙ = 1 + 4 + 4² + … + 4ⁿ⁻¹ = (4ⁿ - 1)/3. Check with n = 2: c₂ = 5, and direct work gives 4x/√(1+45x²), which matches.
4. So. α = c₁₀ = (4¹⁰-1)/3, so 3α + 1 = 4¹⁰ and √(3α+1) = 2¹⁰ = 1024.

The common slip: 1048576. Stopping at 3α + 1 = 4¹⁰ gives 1048576. The question asks for its square root, which is 1024.

The idea to keep. For repeated composition, assume the answer keeps the same shape and find how each constant changes from one step to the next. That recurrence usually turns into a geometric series.

Answer from the NTA final answer key. Solution by PrepWiser.