5.2 Geometric Progressions
A geometric progression is a sequence of numbers where each subsequent term is obtained by multiplying the previous term by a fixed, non-zero constant known as the common ratio, \(r\). Geometric progressions model various real-world phenomena such as bacterial cell division, repeated percentage changes, compound reductions, and layered geometric patterns.
Learning Objectives
- Identify a geometric progression and determine its common ratio, \(r\).
- Derive and apply the formula for the \(n^{\text{th}}\) term, \(T_n\), of a geometric progression.
- Derive and apply the formula for the sum of the first \(n\) terms, \(S_n\), of a geometric progression.
- Determine and apply the formula for the sum to infinity, \(S_\infty\), of a geometric progression where \(|r| < 1\).
- Solve real-world and non-routine problems involving geometric progressions.
Geometric Progression: Term Formula, Sum, and Sum to Infinity
A clear reference roadmap connecting the common ratio, \(n^{\text{th}}\) term formula, both sum formulae, and the condition required for sum to infinity.
Geometric Progression Roadmap
Ratio test: \(r = \dfrac{T_n}{T_{n-1}}\)
\(n^{\text{th}}\) term: \(T_n = a r^{n-1}\)
Sum of first \(n\) terms: \(S_n = \dfrac{a(r^n - 1)}{r - 1}\) for \(|r| > 1\)
Sum to infinity: \(S_\infty = \dfrac{a}{1 - r}\), valid strictly when \(|r| < 1\)
Sum of first \(n\) terms when \(|r| < 1\)
\(S_n = \dfrac{a(1 - r^n)}{1 - r}\)Commonly used when \(-1 < r < 1\).
Sum of first \(n\) terms when \(|r| > 1\)
\(S_n = \dfrac{a(r^n - 1)}{r - 1}\)Commonly used when \(r < -1\) or \(r > 1\).
Short Explanation
A geometric progression begins with a first term, \(a\), and progresses by multiplying each term by a common ratio, \(r\).
1. Identifying a GP
Calculate the ratio between successive terms, \(\dfrac{T_n}{T_{n-1}}\). If this value is constant, the sequence is a GP.
2. Finding the \(n^{\text{th}}\) Term
The power of the common ratio is always one less than the term number, giving \(T_n = a r^{n-1}\).
3. Summing the First \(n\) Terms
When adding \(n\) terms, use \(S_n = \dfrac{a(r^n - 1)}{r - 1}\) for \(|r| > 1\), and \(S_n = \dfrac{a(1 - r^n)}{1 - r}\) for \(|r| < 1\).
4. Sum to Infinity
If \(|r| < 1\), as \(n \to \infty\), \(r^n \to 0\), causing the infinite sum to converge to a finite limit: \(S_\infty = \dfrac{a}{1 - r}\). This formula is useful for converting repeating decimals into exact fractions.
| Concept | Formula / Condition | Notes |
|---|---|---|
| Common ratio | \(r = \dfrac{T_n}{T_{n-1}}\) | Must be constant for a GP. |
| \(n^{\text{th}}\) term | \(T_n = a r^{n-1}\) | Power is \(n-1\), not \(n\). |
| Sum of first \(n\) terms, \(|r| < 1\) | \(S_n = \dfrac{a(1 - r^n)}{1 - r}\) | \(r \neq 1\). |
| Sum of first \(n\) terms, \(|r| > 1\) | \(S_n = \dfrac{a(r^n - 1)}{r - 1}\) | \(r \neq 1\). |
| Term from sums | \(T_n = S_n - S_{n-1}\) | Useful for finding a specific term. |
| Sum to infinity | \(S_\infty = \dfrac{a}{1 - r}\) | Valid only when \(|r| < 1\). |
Converting a Repeating Decimal to a Fraction
Separate the non-repeating part from the repeating part.
Write the repeating part as an infinite geometric series.
Identify \(a\) and \(r\), then apply \(S_\infty = \dfrac{a}{1 - r}\).
Add the non-repeating part to obtain the exact fraction.
Example: \(0.56363\dots = 0.5 + (0.063 + 0.00063 + 0.0000063 + \dots)\)
\(S_\infty = \dfrac{0.063}{1 - 0.01} = \dfrac{7}{110}\)Total fraction: \(\dfrac{1}{2} + \dfrac{7}{110} = \dfrac{31}{55}\).
Try to Answer First
Answer in your mind, then press "Check Answer".
Determine whether the sequence \(120, 40, \dfrac{40}{3}, \dots\) is a geometric progression. Justify your answer.
Given the geometric progression \(4, -20, 100, -500, \dots\), find the common ratio and the \(5^{\text{th}}\) term.
Find the sum to infinity of the geometric progression \(45, 9, 1.8, \dots\).
Common Mistakes
- Inverting the ratio when calculating \(r\), such as writing \(\dfrac{T_1}{T_2}\) instead of \(\dfrac{T_2}{T_1}\).
- Using \(n\) instead of \(n - 1\) in the power of the \(n^{\text{th}}\) term formula, e.g. mistakenly calculating \(a r^n\) instead of \(a r^{n-1}\).
- Forgetting brackets around negative values of \(r\) when raising to powers, e.g. evaluating \(-5^4\) as \(-625\) instead of \((-5)^4 = 625\).
- Applying the sum to infinity formula, \(S_\infty = \dfrac{a}{1 - r}\), when \(|r| \ge 1\), which is invalid because the series diverges.
- In problems involving repeating decimals, taking the initial non-repeating decimal as part of the geometric progression rather than separating it, e.g. treating \(0.5\) as \(a\) in \(0.56363\dots\).
Concept Misunderstandings
Geometric progression graphs are continuous like exponential function curves.
Geometric progression graphs are discrete, defined only at integer positions \(n = 1, 2, 3, \dots\), whereas exponential function curves are continuous across real numbers.
\(S_\infty\) means the sum becomes infinitely large.
When \(|r| < 1\), each term becomes progressively smaller, converging to zero, which means the infinite sum approaches a fixed finite value.
The sum between two specific terms is the same as \(S_n\).
To find the sum of terms from term \(m\) to term \(n\), subtract the sum of the preceding terms: \(S_n - S_{m-1}\).
Summary
- A geometric progression has a constant ratio: \(r = \dfrac{T_n}{T_{n-1}}\).
- The \(n^{\text{th}}\) term is given by \(T_n = a r^{n-1}\).
- The sum of the first \(n\) terms is given by \(S_n = \dfrac{a(1 - r^n)}{1 - r}\) for \(|r| < 1\), and \(S_n = \dfrac{a(r^n - 1)}{r - 1}\) for \(|r| > 1\) (\(r \neq 1\)).
- The \(n^{\text{th}}\) term can also be found via sums: \(T_n = S_n - S_{n-1}\).
- The sum to infinity is valid only when \(|r| < 1\): \(S_\infty = \dfrac{a}{1 - r}\).
Short Activity
Objective: Identify geometric progressions, apply the \(n^{\text{th}}\) term formula, sum formulae and sum to infinity.
A. Objective Quiz
1 Given the geometric progression \(2, \dfrac{2}{3}, \dfrac{2}{9}, \dfrac{2}{27}, \dots\), what is the common ratio \(r\) and the \(7^{\text{th}}\) term \(T_7\)?
2 Under which condition is the sum to infinity, \(S_\infty = \dfrac{a}{1 - r}\), valid for a geometric progression?
B. Fill in the Blanks
3 The \(n^{\text{th}}\) term of a geometric progression with first term \(a\) and common ratio \(r\) is given by the formula \(T_n =\) ________.
4 If the first term is \(a\) and the common ratio is \(r\) where \(|r| < 1\), the sum to infinity formula is \(S_\infty =\) ________.
C. Matching / Drag and Drop
Drag each concept to the correct mathematical expression. On a phone, tap a concept first, then tap the matching box.
