We have mastered Arithmetic Progressions, where terms grow by adding a constant difference. But what if a sequence grows by multiplying?
This creates a Geometric Progression (GP). Instead of a common difference, a GP has a common ratio (\(r\)). Because multiplication compounds upon itself, a GP does not grow in a straight line—it curves and explodes upwards (or shrinks rapidly) in what we call exponential growth.
This creates a Geometric Progression (GP). Instead of a common difference, a GP has a common ratio (\(r\)). Because multiplication compounds upon itself, a GP does not grow in a straight line—it curves and explodes upwards (or shrinks rapidly) in what we call exponential growth.
1 Discovering Multiplicative Patterns
2 Identifying Common Ratios
Select the correct option:
What is the defining characteristic of a Geometric Progression (GP)?
Concept:
1
A sequence \(t_1, t_2, t_3 \dots\) is a Geometric Progression if \(\frac{t_2}{t_1} = \frac{t_3}{t_2} = r\), where \(r\) is the constant common ratio.
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Hint:
1
Does a GP use addition/subtraction, or multiplication/division?
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Mistakes to Avoid:
!
Confusing AP and GP: Arithmetic = Addition. Geometric = Multiplication.
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Solution:
A GP is fundamentally defined by multiplying a term by a constant value to generate the next term, creating a constant ratio between them.
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Select the correct option:
What is the common ratio (\(r\)) of the sequence: \(3, 12, 48, 192 \dots\)?
Concept:
1
Calculate \(r\) by dividing a term by the term before it. \(r = \frac{t_2}{t_1}\).
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Hint:
1
What is \(\frac{12}{3}\)?
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Mistakes to Avoid:
!
Subtracting instead of dividing: If you calculate \(12 - 3 = 9\), you are looking for a common difference, not a common ratio.
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Solution:
\(r = \frac{t_2}{t_1} = \frac{12}{3} = 4\)
Verify: \(12 \times 4 = 48\). The ratio holds.
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Select the correct option:
Which of the following sequences is a Geometric Progression?
Concept:
1
A sequence is a GP if and only if \(\frac{t_2}{t_1} = \frac{t_3}{t_2}\).
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Hint:
1
Test Option C: \(\frac{10}{5} = ?\) and \(\frac{20}{10} = ?\)
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Mistakes to Avoid:
!
Mistaking AP for GP: Option A is a perfectly valid sequence, but it grows by addition (\(+2\)), not multiplication. Therefore it is an AP, not a GP.
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Solution:
Checking Option C: \(5, 10, 20, 40\)
\(\frac{10}{5} = 2\)
\(\frac{20}{10} = 2\)
Since the ratio is constant, it is a valid GP.
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Select the correct option:
If the first term of a GP is \(a=2\), and the common ratio is \(r=3\), what is the 4th term (\(t_4\))?
Concept:
1
To find the terms of a GP, multiply the current term by \(r\) to get the next term.
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Hint:
1
\(t_1 = 2\)
2
\(t_2 = 2 \times 3 = 6\)
3
Keep multiplying by 3 until you reach \(t_4\).
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Mistakes to Avoid:
!
Using AP logic: Do not calculate \(2 + 3(3) = 11\). That is the formula for an AP, not a GP!
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Solution:
Building the GP:
\(t_1 = 2\)
\(t_2 = 2 \times 3 = 6\)
\(t_3 = 6 \times 3 = 18\)
\(t_4 = 18 \times 3 = 54\)
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Select the correct option:
If a Geometric Progression has a common ratio \(r = \frac{1}{2}\) and a positive starting term, what happens to the sequence as it progresses?
Concept:
1
In a GP, if \(r > 1\), the sequence exhibits exponential growth.
2
If \(0 < r < 1\) (a proper fraction), the sequence exhibits exponential decay (shrinking).
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Hint:
1
Imagine starting with 100.
2
Multiply by \(\frac{1}{2}\). You get 50. Then 25. What is happening?
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Mistakes to Avoid:
!
Assuming fractions cause negatives: Repeatedly halving a positive number (like 25 \(\rightarrow\) 12.5 \(\rightarrow\) 6.25 \(\dots\)) will get infinitely close to zero, but will never actually become negative!
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Solution:
Because the common ratio \(\frac{1}{2}\) is less than 1, multiplying any positive term by it will yield a smaller result.
Therefore, the sequence will continually decrease towards zero.
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