Can you predict the number of squares in Stages 5 and 6 of the pattern? In Stages 10,11 and 12? Class 9

Can you predict the number of squares in Stages 5 and 6 of the pattern? In Stages 10,11 and 12? Class 9

Question 1.

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 of squares in AP. Class 9

Answer:

The number of squares at successive stages is 3, 6, 12, 24,...

At each stage, the number of squares is doubled.

So, the number of squares at successive stages can be written as

3, 3 × 2, 3 × 2 × 2, 3 × 2 × 2 × 2, ...

This may be rewritten as 3 × 2⁰, 3 × 2¹, 3 × 2², 3 × 2³,...

So at stage n, the number of squares

= 3 × 2^n-1

By using this formula,

Stage 5 = 3 × 2⁴

Stage 6 = 3 × 2⁵

Stage 10 = 3 × 2⁹

Stage 11 = 3 × 2¹⁰

Stage 12 = 3 × 2¹¹

Stage 20 = 3 × 2¹⁹

∴ At any stage n, the number of squares is given by t_n = 3 × 2^n-1.

Difference:

In the previous growing pattern of squares based on Arithmetic Progression, we added a fixed amount (4) at each stage. Here, we are multiplying by a fixed amount (2) at each stage; this is a Geometric Progression.