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NCERT Solutions for Class 9th Maths Chapter 8 In-text Questions — Fun with Fractals

Book page 189 Updated on2026-09-08

Q1.
The number of black triangles increases very quickly as the stage numbers increase. Can you explain why?
Answer

Because the growth is by multiplication, not addition. Every black triangle turns into three at the next stage, so the whole count triples each time.

tn = 3 × tn–1   gives   tn = 3n
Stage 5: 243    Stage 10: 59,049    Stage 15: 1,43,48,907

Compare this with the AP of Fig. 8.3, where each stage adds a fixed 4:

Stage n51015
AP: 4n – 3173757
GP: 3n24359,0491,43,48,907
Why it happens: when a quantity grows by addition, each step contributes the same fixed amount no matter how large the quantity already is. When it grows by multiplication, each step contributes an amount proportional to what is already there — the bigger it is, the bigger the next jump. In 3n the position n sits in the exponent, and every increase of 1 in n multiplies the whole answer by 3. That compounding is what makes the count run away.
Did you know? this is the same reason a fixed rate of compound interest overtakes a fixed rupee-amount of simple interest once enough time has passed.
Q2.
Can you explain why the area of the black region at Stage n will be (3/4)ⁿ?
Answer

Because the same operation is applied to every black triangle at every stage, and it always keeps exactly three of four equal parts.

One stage of the construction:
divide each black triangle into 4 equal small triangles
remove the middle one, keep 3
black area after the step = (3/4) × black area before the step

Applying this repeatedly from Stage 0, where the area is 1:

s1 = (3/4) × 1 = 3/4
s2 = (3/4) × (3/4) = (3/4)2
s3 = (3/4) × (3/4)2 = (3/4)3
…
sn = (3/4) × sn–1 = (3/4)n
Why it happens: joining the midpoints of a triangle cuts it into four triangles of equal area, so removing one removes exactly one quarter — whatever the size of the triangle. That is what lets the step be applied simultaneously to all 3n black triangles at a stage and still be described by the single factor 3/4. Multiplying by 3/4 exactly n times, starting from 1, is the meaning of (3/4)n. This is the recursive rule and the explicit rule for the same GP, and the argument shows why they agree.
Check it yourself: at Stage n there are 3n black triangles, each of area (1/4)n. Total area = 3n × (1/4)n = (3/4)n — the same answer by a different route.
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