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.
A clear reference roadmap connecting the common ratio, \(n^{\text{th}}\) term formula, both sum formulae, and the condition required for sum to infinity.
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 \(|r| < 1\)
Commonly used when \(-1 < r < 1\).
Sum of first \(n\) terms when \(|r| > 1\)
Commonly used when \(r < -1\) or \(r > 1\).
A geometric progression begins with a first term, \(a\), and progresses by multiplying each term by a common ratio, \(r\).
Calculate the ratio between successive terms, \(\dfrac{T_n}{T_{n-1}}\). If this value is constant, the sequence is a GP.
The power of the common ratio is always one less than the term number, giving \(T_n = a r^{n-1}\).
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 \(|r| < 1\).
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.
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: \(0.56363\dots = 0.5 + (0.063 + 0.00063 + 0.0000063 + \dots)\)
Total fraction: \(\dfrac{1}{2} + \dfrac{7}{110} = \dfrac{31}{55}\).
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 \(45, 9, 1.8, \dots\).
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: \(S_n - S_{m-1}\).
Objective: Identify geometric progressions, apply the \(n^{\text{th}}\) term formula, sum formulae and sum to infinity.
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?
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 \(S_\infty =\) ________.
Drag each concept to the correct mathematical expression. On a phone, tap a concept first, then tap the matching box.
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