Welcome to **Exercise 8.3**! Having explored how geometric progressions scale exponentially through fractals and bouncing physics, we will now master the core textbook problems on Page 193.
In this part, we compute common ratios, generate explicit geometric terms, and build foundational fluency with exponential scaling. Let's solve!
In this part, we compute common ratios, generate explicit geometric terms, and build foundational fluency with exponential scaling. Let's solve!
1 Generating GP Terms
2 Identifying Specific Positions in a GP
Select the correct option:
What is the explicit formula for the \(n^{th}\) term (\(t_n\)) of a Geometric Progression?
Concept:
1
To find the \(n^{th}\) term of a GP, multiply the first term \(a\) by the common ratio \(r\) raised to the power of \((n-1)\).
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Hint:
1
For the 1st term (\(n=1\)), the exponent must be \(1-1 = 0\), making \(r^0 = 1\), so \(t_1 = a\).
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Mistakes to Avoid:
!
Using exponent \(n\): If you use \(a \cdot r^n\), for \(n=1\) you get \(a \cdot r\), which skips multiplying by the first term correctly.
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Solution:
The general explicit term of a GP is \(t_n = a \cdot r^{n-1}\).
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Select the correct option:
What is the common ratio (\(r\)) of the GP: \(5, 1, \frac{1}{5}, \frac{1}{25}, \dots\)?
Concept:
1
Calculate the common ratio by dividing the second term by the first term: \(r = \frac{t_2}{t_1}\).
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Hint:
1
Divide \(\frac{1}{5}\) by \(5\).
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Mistakes to Avoid:
!
Flipping the fraction: Do not write 5. Dividing 1 by 5 gives 1/5, whereas dividing 5 by 1 gives 5. Always put the second term on top!
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Solution:
\(r = \frac{t_2}{t_1} = \frac{1}{5}\).
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Select the correct option:
If the first term of a GP is \(a = 3\) and the common ratio is \(r = 2\), what is the 5th term (\(t_5\))?
Concept:
1
Use \(t_5 = a \cdot r^4\).
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Hint:
1
Calculate \(3 \times 2^4 = 3 \times 16\).
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Mistakes to Avoid:
!
Using exponent 5: The power is \((n-1)\), so for the 5th term, the exponent is 4.
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Solution:
\(t_5 = 3 \cdot 2^4 = 3 \times 16 = 48\).
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Select the correct option:
Which of the following represents a valid Geometric Progression?
Concept:
1
Test consecutive terms for a constant division ratio.
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Hint:
1
Check Option B: \(\frac{6}{2} = 3\), \(\frac{18}{6} = 3\), \(\frac{54}{18} = 3\).
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Mistakes to Avoid:
!
Confusing AP with GP: Option A grows by adding 3, which is arithmetic, not geometric.
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Solution:
Option B features a constant ratio of 3 across all consecutive terms, making it a valid GP.
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Select the correct option:
If a GP has a common ratio \(r = -2\) and starting term \(a = 3\), what is the 3rd term (\(t_3\))?
Concept:
1
Negative ratios cause the sequence to alternate signs (+, -, +, -).
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Hint:
1
\(t_3 = 3 \cdot (-2)^2 = 3 \cdot 4 = 12\).
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Mistakes to Avoid:
!
Squaring negative numbers incorrectly: \((-2)^2 = +4\). Multiplying a negative by a negative yields a positive!
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Solution:
\(t_3 = 3 \cdot (-2)^{3-1} = 3 \cdot (-2)^2 = 3 \times 4 = 12\).
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