Definition and the general term
Learning Objectives
- State the definition of a geometric progression and identify the common ratio.
- Prove and apply the closed-form formula for the \(n\)-th term.
- Recognise the geometric-mean characterisation of a geometric progression.
- Relate the magnitude and sign of the common ratio to monotonicity and boundedness.
Motivation
Many situations of change are not additive but multiplicative. A population that doubles every hour, a savings account whose balance grows by a fixed percentage each year, a piece of paper whose thickness doubles with every fold — at each step the quantity is multiplied by the same factor, not increased by the same amount. The structural form of these situations is different from the arithmetic progressions of the previous unit, and it forces us to introduce a new family.
A famous illustration is the chessboard wheat legend, told in several mediaeval traditions about the inventor of the game of chess. The inventor asks his ruler for a modest reward: one grain of wheat on the first square of the board, two on the second, four on the third, and so on, doubling each time. The king, expecting a small request, agrees. By the sixty-fourth square the required count has reached \(2^{63}\) grains — more wheat than has ever been grown on Earth. The story turns on the fact that multiplicative growth, even at a modest rate, eventually outstrips any linear comparison.
Geometric progression
A sequence \((b_n)_{n \geq 1}\) of real numbers, with \(b_1 \neq 0\), is called a geometric progression if there exists a real number \(q \neq 0\) such that
\[b_{n+1} = q \cdot b_n \quad \text{for every } n \geq 1.\]
The number \(q\) is called the common ratio of the progression.
Equivalently, for a geometric progression the ratio of consecutive terms is constant: \(\dfrac{b_{n+1}}{b_n} = q\) for every \(n \geq 1\), which is the source of the name. The conditions \(b_1 \neq 0\) and \(q \neq 0\) ensure that every term is nonzero, so that these ratios are well defined throughout.
A geometric progression is determined by two pieces of data: its first term \(b_1\) and its common ratio \(q\). Once these are fixed, every other term is forced.
First examples
\(1,\ 2,\ 4,\ 8,\ 16,\ \ldots\) is a geometric progression with \(b_1 = 1\) and common ratio \(q = 2\) — the powers of two.
\(100,\ 50,\ 25,\ 12.5,\ 6.25,\ \ldots\) is a geometric progression with \(b_1 = 100\) and \(q = \dfrac{1}{2}\). The terms shrink toward zero but never reach it.
\(5,\ 5,\ 5,\ 5,\ \ldots\) is a geometric progression with \(b_1 = 5\) and \(q = 1\). Constant nonzero sequences satisfy the defining condition with \(q = 1\).
\(3,\ -6,\ 12,\ -24,\ 48,\ \ldots\) is a geometric progression with \(b_1 = 3\) and \(q = -2\). Negative ratios produce alternating signs.
\(1,\ 3,\ 5,\ 7,\ 9,\ \ldots\) is not a geometric progression. The ratio \(b_2 / b_1 = 3\) but the ratio \(b_3 / b_2 = 5/3\), so the consecutive ratios are not constant. (It is, of course, an arithmetic progression.)
The general term
Let \((b_n)_{n \geq 1}\) be a geometric progression with first term \(b_1\) and common ratio \(q\). Then, for every \(n \geq 1\),
\[b_n = b_1 \cdot q^{n-1}.\]
We proceed by induction on \(n \geq 1\).
Base case. When \(n = 1\), the formula reads \(b_1 = b_1 \cdot q^{0} = b_1 \cdot 1 = b_1\), which holds.
Inductive step. Suppose \(b_n = b_1 \cdot q^{n-1}\) for some \(n \geq 1\). By the defining condition of a geometric progression, \(b_{n+1} = q \cdot b_n\). Substituting the inductive hypothesis,
\[b_{n+1} = q \cdot \bigl(b_1 \cdot q^{n-1}\bigr) = b_1 \cdot q^{n} = b_1 \cdot q^{(n+1)-1}.\]
This is the formula at index \(n + 1\), completing the induction.
By the Principle of Mathematical Induction, the formula \(b_n = b_1 \cdot q^{n-1}\) holds for every \(n \geq 1\).
A note on indexing
As with arithmetic progressions, the convention used here is to start the index at \(n = 1\). With indexing from \(n = 0\), the same progression is written
\[b_n = b_0 \cdot q^{n} \quad \text{for } n \geq 0,\]
which has the same shape but a slightly cleaner exponent. The mathematics is the same; only the label attached to each term shifts by one. In computer science the zero-indexed form is more common, since exponents and array indices then agree.
Monotonicity and boundedness from the ratio
The behaviour of a geometric progression is governed by two features of its common ratio: its sign and its magnitude. The combined picture is richer than for arithmetic progressions, where only the sign of the common difference mattered.
Sign of \(q\). If \(q > 0\) the terms all share the sign of \(b_1\). If \(q < 0\) the terms alternate in sign and the sequence is not monotone, no matter what \(b_1\) is.
Magnitude of \(q\) relative to \(1\). Under the assumption \(q > 0\), the progression is strictly increasing in absolute value when \(q > 1\), constant when \(q = 1\), and strictly decreasing in absolute value when \(0 < q < 1\). For \(b_1 > 0\) these statements become statements about the terms themselves; for \(b_1 < 0\) the direction reverses.
Boundedness. A geometric progression is bounded if and only if \(|q| \leq 1\). When \(|q| < 1\) the absolute values shrink toward zero; when \(|q| = 1\) the terms have constant absolute value \(|b_1|\); when \(|q| > 1\) the absolute values grow beyond every bound. This last case is what powers the chessboard story: with \(q = 2\), the terms outgrow any imaginable scale within a few dozen steps.
The geometric-mean characterisation
A sequence \((b_n)_{n \geq 1}\) of nonzero real numbers is a geometric progression if and only if, for every \(n \geq 2\),
\[b_n^{\,2} = b_{n-1} \cdot b_{n+1}.\]
When the terms are all positive, this is equivalent to \(b_n = \sqrt{b_{n-1} \cdot b_{n+1}}\): each term is the geometric mean of its two neighbours. This is the property from which the family takes its name, just as the arithmetic-mean property gave the name to arithmetic progressions.
(⇒) Suppose \((b_n)\) is a geometric progression with common ratio \(q\). Then \(b_{n+1} = q \cdot b_n\) and \(b_{n-1} = b_n / q\), so
\[b_{n-1} \cdot b_{n+1} = \frac{b_n}{q} \cdot (q \cdot b_n) = b_n^{\,2}.\]
(⇐) Suppose \(b_n^{\,2} = b_{n-1} \cdot b_{n+1}\) for every \(n \geq 2\), with every \(b_n\) nonzero. Dividing both sides of this identity by \(b_{n-1} \cdot b_n\) (both nonzero), we obtain
\[\frac{b_n}{b_{n-1}} = \frac{b_{n+1}}{b_n} \quad \text{for every } n \geq 2.\]
The ratio of consecutive terms is therefore the same at every position; call it \(q\). Then \(q = b_2 / b_1\), and \(b_{n+1} = q \cdot b_n\) for every \(n \geq 1\), which is the defining condition of a geometric progression.
Using the general term
Suppose a geometric progression has \(b_1 = 5\) and \(q = 3\). The general term is
\[b_n = 5 \cdot 3^{\,n-1}.\]
So \(b_4 = 5 \cdot 27 = 135\), and \(b_{10} = 5 \cdot 3^{9} = 5 \cdot 19683 = 98415\). The contrast with the arithmetic case is striking: ten steps of a modest geometric progression already produce a five-figure term.
Conversely, suppose we are told that a geometric progression has \(b_3 = 12\) and \(b_6 = 96\), and asked to recover \(b_1\) and \(q\). From the general term, \(b_6 / b_3 = q^{3}\), so
\[q^{3} = \frac{96}{12} = 8 \quad \Longrightarrow \quad q = 2.\]
Then \(b_3 = b_1 \cdot q^{2} = 4 b_1 = 12\), so \(b_1 = 3\). The progression is \(3,\ 6,\ 12,\ 24,\ 48,\ \ldots\)
Connection to Computer Science
Geometric progressions are the natural model for any process in which work or quantity is multiplied at each step rather than added. A binary tree of depth \(n\) has \(2^{n}\) leaves; a dataset that doubles every year fills a fixed storage capacity after a predictable number of years; a recursive procedure that halves its input at every call needs only \(\log_2 n\) levels of recursion to reach the base case. In each example a geometric progression is in the background, with common ratio \(2\) or \(1/2\).
The same structure produces the contrast between linear and exponential growth that animates much of algorithm analysis. A loop that performs the same fixed amount of work per iteration is linear in the input size; one that doubles its work at each step is exponential, and quickly becomes infeasible. Recognising whether a recurrence describes an arithmetic or a geometric progression is often the key to estimating how a program will scale — a question the closed-form general term answers in a single calculation, without simulating any of the intermediate steps.
Why exclude \(b_1 = 0\) and \(q = 0\)?
The definition requires both the first term and the common ratio to be nonzero. The reason is technical but worth examining.
If \(b_1 = 0\), then every subsequent term \(b_n = b_1 \cdot q^{n-1}\) is also zero, regardless of \(q\). The common ratio is no longer determined by the progression: \(q\) could be any nonzero number and the sequence would look identical. Allowing \(b_1 = 0\) would therefore make the common ratio meaningless.
If \(q = 0\) (with \(b_1 \neq 0\)), then \(b_2 = 0, b_3 = 0,\) and so on. The first term is nonzero but every subsequent term vanishes. The ratio \(b_{n+1}/b_n\) becomes \(0/0\) starting at \(n = 2\), and the defining condition \(b_{n+1} = q \cdot b_n\) degenerates. To keep ratios always well defined and the progression always nontrivial, the case \(q = 0\) is excluded.
With these conventions in place, every geometric progression has nonzero terms throughout, and dividing by any term — as we did in the proof of the geometric-mean characterisation — is always legitimate.
Exercises
Exercise 1
For each sequence, decide whether it is a geometric progression.
Exercise 2
A geometric progression has \(b_1 = 3\) and common ratio \(q = 2\). Use the general term formula to fill in the requested values.
Exercise 3
A geometric progression has \(b_3 = 18\) and \(b_6 = 486\). Find its common ratio and first term.
Exercise 4
Decide whether each statement is true or false.
Summary
A geometric progression is a sequence \((b_n)_{n \geq 1}\) with \(b_1 \neq 0\) and a fixed nonzero common ratio \(q\), meaning \(b_{n+1} = q \cdot b_n\) for every \(n\).
The progression is determined by \(b_1\) and \(q\), and its general term is
\[b_n = b_1 \cdot q^{n-1}.\]
Behaviour depends on the sign and magnitude of \(q\). A positive ratio preserves sign; a negative ratio forces alternation. A ratio with \(|q| < 1\) shrinks the terms toward zero (bounded); \(|q| = 1\) keeps absolute values constant (also bounded); \(|q| > 1\) sends the terms beyond every bound (unbounded).
A sequence of nonzero real numbers is a geometric progression if and only if \(b_n^{\,2} = b_{n-1} \cdot b_{n+1}\) for every \(n \geq 2\) — the geometric-mean characterisation, from which the family takes its name.
Geometric progressions are the discrete analogue of exponential functions, just as arithmetic progressions are the discrete analogue of linear ones.