The number of black triangles at the nth stage of the Sierpiński triangle is given by 3n. The number of black triangles increases very quickly as the stage numbers increase. Can you explain why?
Step-by-step solution
Idea: Growth by multiplying (a GP with ratio more than 1) is much faster than growth by adding (an AP), because each stage builds on all the triangles already there.
- At each stage, every black triangle is replaced by 3 smaller black triangles. So the count is multiplied by 3: it forms a GP with common ratio 3.1 mark
- The more triangles there are, the more get added next time: 3 → 9 adds 6, 27 → 81 adds 54, 2187 → 6561 adds 4374. After 10 stages there are 310 = 59049, whereas adding even 3 at each stage would give only 31.1 mark
Answer to write in the exam
Each black triangle → 3 black triangles at the next stage
∴ Number of triangles is multiplied by 3 at each stage (GP, r = 3)
1, 3, 9, 27, 81, … ; Stage 10 = 310 = 59049
∴ Repeated multiplication by 3 makes the number increase very quickly.
Common mistakes that cost marks
- Saying ‘three triangles are added each stage’. Each triangle becomes three, so the total triples.
- Thinking the triangles grow because the area grows. The area actually shrinks; only the number of (smaller) triangles grows.
- Comparing with the wrong kind of sequence: an AP adds the same amount; a GP with r > 1 multiplies.
How this can come in the exam
Assertion (A): The Sierpiński triangle has 729 black triangles at Stage 6.
Reason (R): The number of black triangles is multiplied by 3 at each stage, starting from 1 at Stage 0.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
36 = 729, and R is exactly why.
Try one yourself
A rumour is told to 1 person on day 0, and each day every person who knows it tells 4 new people. How many new people hear it on day 5? Why does it spread so quickly?
Show answer
The number who know it is multiplied by 5 each day: 1, 5, 25, 125, 625, 3125. New hearers on day 5 = 3125 − 625 = 2500. It spreads fast because everyone who knows it passes it on, so the numbers multiply.
More questions like this
- If the black region at Stage 0 is 1 square unit, then the black region at Stage 1 is 34 square units. This process is repeated on Stage 1 to arrive at Stage 2. Hence, the black region at Stage 2 will be 34 of the black region of Stage 1, which is equal to 34 × 34 = (34)2. Can you explain why the area of the black region at Stage n will be (34)n?
- A ball is dropped from a height of 24 feet above the ground. Each time the ball bounces up to (34)th of its previous height.
- Find the 12th term of a GP with common ratio 2, whose 8th term is 192.
- Find the 10th and nth terms of the GP: 5, 25, 125, … .
- A sequence is given by the recursive rule t1 = 2, tn+1 = 3tn − 2 for n ≥ 1. Which term of the sequence is 730?
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