Q1.
Observe the Sierpiński triangle and try to answer the following questions. (a) How many black triangles are there in Stages 0 to 3 of Fig. 8.7? (b) Can you predict the number of black triangles at Stages 4 and 5? (c) Can you find a rule for the number of black triangles at the nth stage? (d) Suppose the area of the triangle (that is, the black region) in Stage 0 is 1 square unit. What is the area of the black region in Stages 1, 2 and 3? What will be the area of the black region in Stages 4 and 5? Find a rule for the area of the black region at the nth stage. What happens to this area as n, the number of stages, goes on increasing?
Answer
(a) Counting the black triangles in Fig. 8.7:
Stage 0: 1 Stage 1: 3 Stage 2: 9 Stage 3: 27
(b) Each black triangle is replaced by 3 smaller black triangles at the next stage, so the count is multiplied by 3.
Stage 4: 27 × 3 = 81 Stage 5: 81 × 3 = 243
(c) The counts 1, 3, 9, 27, 81, 243, … are a GP with first term 1 and common ratio 3, and they are the powers of 3 with the exponent equal to the stage number:
1 = 30, 3 = 31, 9 = 32, 27 = 33, 81 = 34, 243 = 35
Number of black triangles at Stage n = 3n
Recursively: t0 = 1, tn = 3 × tn–1
Number of black triangles at Stage n = 3n
Recursively: t0 = 1, tn = 3 × tn–1
(d) At each stage the triangle is cut into 4 equal parts and the middle one is removed, so exactly 3/4 of the black area survives.
Stage 1: 1 × 3/4 = 3/4 = 0.75
Stage 2: (3/4) × (3/4) = 9/16 = 0.5625
Stage 3: (3/4)3 = 27/64 ≈ 0.4219
Stage 4: (3/4)4 = 81/256 ≈ 0.3164
Stage 5: (3/4)5 = 243/1024 ≈ 0.2373
Area at Stage n = (3/4)n square units
Recursively: s0 = 1, sn = (3/4) × sn–1
Stage 2: (3/4) × (3/4) = 9/16 = 0.5625
Stage 3: (3/4)3 = 27/64 ≈ 0.4219
Stage 4: (3/4)4 = 81/256 ≈ 0.3164
Stage 5: (3/4)5 = 243/1024 ≈ 0.2373
Area at Stage n = (3/4)n square units
Recursively: s0 = 1, sn = (3/4) × sn–1
As n increases the area keeps being multiplied by 3/4, a number less than 1, so the black area shrinks steadily and creeps closer and closer to 0 without ever becoming 0.
Why it happens: the same figure produces two GPs pulling in opposite directions. Counting triangles multiplies by 3 (r > 1, so the count explodes); measuring area multiplies by 3/4 (0 < r < 1, so the area collapses). There is no contradiction: at each stage there are three times as many pieces, but each piece has only a quarter of the area of the piece it came from, and 3 × 1/4 = 3/4 < 1. That single number, 3/4, is why the count of pieces and the total area move opposite ways.
Did you know? the numbering starts at Stage 0, so here the exponent equals the stage number exactly — 3n and (3/4)n, with no "n – 1". Always check what the first stage is called before writing a formula.