Let f: ℝ → ℝ be a continuous function satisfying f(0)=1 and f(2 x)-f(x)=x for all x ∈ ℝ. If lim n → ∞{f(x)-f(x/2ⁿ)}=G(x), then Σr=1¹⁰ G(r²) is equal to

Solution

Maths sir · 4 steps

Halve x again and again, and the differences telescope into a geometric series that sums to x.

1. Use the equation repeatedly. Put x/2 in place of x to get f(x) - f(x/2) = x/2. In the same way, f(x/2) - f(x/4) = x/4, and this continues down to f(x/2ⁿ⁻¹) - f(x/2ⁿ) = x/2ⁿ.
2. Add and telescope. Adding all n lines cancels every middle term. Only the first and last terms of f survive, and the right sides form a geometric series.

   f(x) - f(x/2ⁿ) = x(1/2 + 1/4 + … + 1/2ⁿ) = x(1 - 1/2ⁿ)

3. Take the limit. As n → ∞, 1/2ⁿ → 0, so G(x) = x. Since f is continuous, this also gives f(x) - f(0) = x, that is, f(x) = x + 1.
4. So. Σᵣ₌₁¹⁰ G(r²) = Σᵣ₌₁¹⁰ r² = (10 × 11 × 21)/6 = 385, which is option (c).

The idea to keep. When a functional equation links f(2x) and f(x), keep halving the argument and add the equations. The middle terms cancel and continuity takes you to f(0).

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