Learnify Academy is a tuition centre in Bahrain. Classes are for students in Bahrain only.Tuition classes in Bahrain only

Geometric progressions · 4 marks

Consider the growing pattern of squares shown in the figure. The first four stages of the pattern are shown. If we count the total number of green squares in the four stages of the pattern, we get the sequence 3, 6, 12, 24. Can you predict the number of squares in Stages 5 and 6 of the pattern? In Stages 10, 11 and 12? In Stage 20? At any stage? How is this different from the growing pattern in the figure?

Pattern of green squares: 3, 6, 12, 24Stage 1Stage 2Stage 3Stage 4Earlier pattern of squares: 1, 5, 9, 13Stage 1Stage 2Stage 3Stage 4
Answer: Stages 5, 6: 48, 96; Stages 10, 11, 12: 1536, 3072, 6144; Stage 20: 1572864; Stage n: 3 × 2n−1. Here the number doubles each stage (multiply by 2), while the earlier pattern 1, 5, 9, 13 adds 4 each stage, so this one grows far faster.

Step-by-step solution

Idea: Each stage has twice as many squares as the stage before. So stage n is 3 doubled (n − 1) times. The earlier pattern added a fixed number instead.

  1. 3, 6, 12, 24: each term is 2 × the previous term (the number of rows doubles, 3 squares per row).½ mark
  2. Stage 5: 24 × 2 = 48; Stage 6: 48 × 2 = 96.½ mark
  3. Stage n: 3 × 2 × 2 × … (n − 1 twos) = 3 × 2n−1.1 mark
  4. Stage 10: 3 × 29 = 3 × 512 = 1536; Stage 11: 3072; Stage 12: 6144. Stage 20: 3 × 219 = 3 × 524288 = 1572864.1 mark
  5. Difference: the earlier pattern 1, 5, 9, 13, … adds the same 4 squares each stage (an AP, 4n − 3; stage 20 has only 77). This pattern multiplies by the same 2 each stage, so it grows much faster (stage 20 has 1572864).1 mark
48 and 96; 1536, 3072 and 6144; 1572864; 3 × 2ⁿ⁻¹ at stage n. This pattern doubles each time (common ratio 2), while the earlier one adds 4 each time (common difference 4), so this one grows much faster.

Check: Stage 4: 3 × 2³ = 24 ✓. Stage 11 = 2 × Stage 10 = 2 × 1536 = 3072 ✓.

Answer to write in the exam

3, 6, 12, 24: each term = 2 × previous term

Stage 5 = 48, Stage 6 = 96

Stage n = 3 × 2n−1

Stage 10 = 3 × 29 = 1536, Stage 11 = 3072, Stage 12 = 6144

Stage 20 = 3 × 219 = 1572864

Earlier pattern: adds 4 each stage (4n − 3); this pattern: multiplies by 2 each stage

∴ This pattern grows much faster (stage 20: 1572864 against 77).

Common mistakes that cost marks

  • Adding 3 each time (3, 6, 9, 12, …). Look again: 6 → 12 → 24 doubles; it does not add a fixed amount.
  • Writing stage n as 3 × 2n (giving 6 at stage 1). Stage 1 has 3 squares, so the power is n − 1.
  • Writing 3 × 219 as 619. Work out 219 first, then multiply by 3.

How this can come in the exam

MCQ (1 mark)

A pattern has 5, 10, 20, 40, … tiles at stages 1, 2, 3, 4. The number of tiles at stage 8 is

  1. 320
  2. 640
  3. 1280
  4. 80
Show answer

(B) 640
5 × 27 = 5 × 128 = 640.

Assertion–Reason (1 mark)

Assertion (A): By stage 10, a pattern with 3, 6, 12, … squares has more squares than a pattern with 1, 5, 9, … squares.
Reason (R): A pattern that multiplies by 2 at every stage eventually grows faster than one that adds a fixed number.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
Show answer

(A) Both A and R are true, and R is the correct explanation of A.
Stage 10: 1536 against 37. R explains A.

Try one yourself

A pattern has 2, 6, 18, 54, … squares at stages 1, 2, 3, 4. Find the number at stages 6 and n.

Show answer

Multiply by 3 each time: stage n = 2 × 3n−1; stage 6 = 2 × 243 = 486.

More questions like this

All Sequences and progressions questions · All maths questions

← A child arranges marbles in rows so that the first row has 1 marble, tIs 1, 2, 4, 8, 16, … a geometric progression? If so, what is the commo →