Question 1 - Think & Reflect (Page 186) - Geometric Progressions - Chapter 8 - Predicting What Comes Next: Exploring Sequences & Progress

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Question 1 - Think & Reflect (Page 186) 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 Fig. 8.3? In this pattern, the squares go: 3, 6, 12, 24, … Here, First term = a = 3 Common ratio = r = 6/3 = 2 Here, First term = a = 3 Common ratio = r = 6/3 = 2 Now, we know that The general formula for nth term in a GP is: 𝒕_𝒏=𝒂 × 𝒓^(𝒏−𝟏) For our specific sequence, the formula is: 𝒕_𝒏=𝟑 × 𝟐^(𝒏−𝟏) Now we can plug in the stage numbers (𝑛) to find the answers: Stage 5: 3 × 2^((5−1))=3×2^4=3×16=𝟒𝟖 Stage 6: 3 × 2^((6−1))=3×2^5=3×32=𝟗𝟔 Stage 10: 3 × 2^((10−1))=3×2^9=3×512=𝟏𝟓𝟑𝟔 Stage 11: 3 × 2^((11−1))=3×2^10=3×1024=𝟑𝟎𝟕𝟐 Stage 12: 3 × 2^((12−1))=3×2^11=3×2048=𝟔𝟏𝟒𝟒 Stage 20: 3 × 2^((20−1))=3×2^19=3×524,288=𝟏,𝟓𝟕𝟐,𝟖𝟔𝟒 And, At any stage 𝑛: 𝟑 × 𝟐^(𝒏−𝟏)