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TMUA 2023 · Paper 1 · Question 17 of 20

TMUA 2023 Paper 1 Question 17

Sequences and series — Circles and series · summing annular areas. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 1Sequences and seriesCircles and series · summing annular areas6 optionshard

The original question includes a diagram: Nested circles C1 to C4 about a common diagonal, with the regions between C1 and C2 and between C3 and C4 shaded.

A circle Cn is defined by x2 + y2 = 2n(x + y) where n is a positive integer.

C1 and C2 are drawn and the area between them is shaded. Next, C3 and C4 are drawn and the area between them is shaded. This process continues until 100 circles have been drawn.

What is the total shaded area?

  1. A100π
  2. B500π
  3. C2500π
  4. D5050π
  5. E10100π
  6. F40400π
Show the answer and worked solution
answer · E
  1. A100π
  2. B500π
  3. C2500π
  4. D5050π
  5. E10100π
  6. F40400π
Completing the square, (x−n)2 + (y−n)2 = 2n2, so Cn has radius n√2 and area 2π n2. The shaded regions are those between C2m−1 and C2m, of area 2π[(2m)2 − (2m−1)2] = 2π(4m−1). With 100 circles there are 50 such regions, so the total is 2π∑m=150(4m−1) = 2π(4⋅50⋅ 512 − 50) = 2π(5050) = 10100π.