Introduction
A series can be difficult to recognize when its terms are written out one by one. A useful first question is whether each term is obtained from the previous term by multiplying by the same number. When that happens, the terms form a geometric sequence, and their sum is called a geometric series.
The common ratio can be positive, negative, or zero. The starting term also need not be \(1\). Recognizing these features is a matter of checking the relationship between consecutive terms, rather than relying on how familiar the series looks.
Definition and notation
A sequence \((a_n)\) is geometric if there is a fixed number \(r\) such that each term is \(r\) times the preceding term. In recurrence form, this means \(a_{n+1}=r a_n\) for every \(n\geq 1\). The number \(r\) is called the common ratio, and \(a_1\) is the starting term.
The exponent \(n-1\) reflects the indexing: when \(n=1\), the exponent is zero, so the first term is \(a_1r^0=a_1\). This representation describes the terms; recognizing a series as geometric does not by itself determine whether the infinite series converges.
If the terms being compared are nonzero, the recurrence can be tested with consecutive quotients:
The same value must occur at every step. A constant ratio is different from a constant difference: adding the same amount each time does not, in general, produce a geometric sequence.
Worked Example: A geometric series with alternating signs
Worked Example: A geometric series with alternating signs
Determine whether the terms \(6,-3,\frac{3}{2},-\frac{3}{4},\ldots\) are geometric. If they are, identify the starting term and common ratio.
Divide each term by the preceding term:
Each quotient is \(-\frac12\), so the sequence is geometric with \(a_1=6\) and \(r=-\frac12\). Its general term is \(a_n=6\left(-\frac12\right)^{n-1}\).
Check: Substituting \(n=1,2,3,4\) gives \(6,-3,\frac32,-\frac34\), respectively. Each term is also \(-\frac12\) times the term before it, as required.
A negative ratio reverses the sign at each step when the terms are nonzero. It is not a reason to reject the geometric pattern: the sign change is part of the fixed multiplication rule.
Worked Example: A nonunit starting term
Worked Example: A nonunit starting term
Consider the series \(15+5+\frac53+\frac59+\cdots\). Check whether it is geometric and write its term pattern using \(a_1\) and \(r\).
The consecutive ratios are
The ratios agree, so the series is geometric with \(a_1=15\) and \(r=\frac13\). Its terms have the form \(15\left(\frac13\right)^{n-1}\).
Check: For \(n=1\), this gives \(15\). For \(n=2\), it gives \(15\cdot\frac13=5\); for \(n=3\), it gives \(15\cdot\frac19=\frac53\); and for \(n=4\), it gives \(15\cdot\frac1{27}=\frac59\). These match the listed terms.
The first term need not be \(1\), and it need not be built into the ratio. The ratio describes how terms change from one index to the next; the starting term sets their initial scale.
Why the ratio test gives the geometric form
The quotient test is useful because it is exactly the recurrence defining a geometric sequence, provided the denominator terms are nonzero. It also explains why the power formula has the exponent \(n-1\).
Proof. The quotient condition gives \(a_{n+1}=r a_n\) after multiplying both sides by \(a_n\). Starting from \(a_1\), apply that relation repeatedly:
More generally, induction proves the stated formula. It holds for \(n=1\), since \(a_1=a_1r^0\). If it holds for some \(n\), then
Thus it holds for \(n+1\), and therefore for every \(n\geq 1\). For the converse, divide consecutive terms using the proposed formula:
Here the division is valid because \(a_1\neq 0\) and \(r\neq 0\). This completes the proof.
Worked Example: Constant differences are not constant ratios
Worked Example: Constant differences are not constant ratios
Decide whether \(2,5,8,11,\ldots\) is geometric. A quick visual pattern is that each term increases by \(3\), but that alone does not answer the ratio question.
Calculate the consecutive ratios:
They are not equal: for example, \(\frac52\neq\frac85\), because \(25\neq16\) after cross-multiplication. Therefore the sequence is not geometric.
Check: If a fixed ratio \(r\) produced the sequence, the first quotient would require \(r=\frac52\), while the second would require \(r=\frac85\). Since those requirements conflict, no single common ratio generates these terms.
Using the recurrence when a quotient is undefined
The quotient method requires division by the preceding term. If that term is zero, the quotient is undefined, so it cannot be used at that step. The recurrence definition \(a_{n+1}=r a_n\) remains available and is the safer test when zeros occur.
Worked Example: A zero common ratio
Consider the terms \(9,0,0,0,\ldots\). The quotient \(0/9\) is defined and equals \(0\), but the next quotient \(0/0\) is undefined. Does that prevent the sequence from being geometric?
Use the recurrence instead. With \(a_1=9\) and \(r=0\), the next term must be \(a_2=0\cdot9=0\). Every later term must also be zero, since \(0\cdot0=0\). Thus the sequence satisfies \(a_{n+1}=0a_n\) for every \(n\), and it is geometric with common ratio \(0\).
Check: The recurrence produces \(9,0,0,0,\ldots\), exactly the listed terms. The undefined quotient \(0/0\) is not evidence against the pattern; it only shows that quotient testing is unavailable there.
If the first term is zero, the recurrence forces every later term to be zero for any fixed \(r\). The all-zero sequence therefore satisfies the recurrence, although consecutive quotients cannot identify a unique ratio. For any sequence containing zeros, state the recurrence you are checking rather than treating an undefined quotient as a numerical ratio.
A reliable recognition procedure
For a sequence whose consecutive terms are nonzero, checking the first few quotients is a practical way to look for a geometric pattern. To establish the pattern for an entire sequence, the same quotient must be shown to hold for every relevant index, not merely for a few displayed terms. A formula for the terms can make that verification possible.
Read the series as a sequence \(a_1,a_2,\ldots\), paying attention to which term is first.
When \(a_n\neq0\), compute \(a_{n+1}/a_n\). Do not compare nonconsecutive terms as if they were adjacent.
If the quotients differ, the sequence is not geometric. If zeros make a quotient undefined, test the recurrence \(a_{n+1}=ra_n\) instead.
For a geometric sequence, state \(a_1\) and \(r\), and write \(a_n=a_1r^{n-1}\) when the indexing begins at \(n=1\).
Substitute early indices into the proposed expression, or multiply each term by \(r\), and confirm that the original terms result.
Common pitfall: using a ratio that skips terms
A common error is to compare a term with the one two places earlier, then report that quotient as the common ratio. That quotient represents two multiplication steps, not one.
The correction is to divide adjacent terms:
The common ratio is \(3\), not \(9\). The skipped-term quotient is \(9=3^2\), because two multiplications by the adjacent ratio take the sequence from \(4\) to \(36\). Checking adjacency prevents this indexing mistake.
Why recognition matters
Recognizing a geometric pattern gives a compact description of a series: instead of listing every term, one can specify a starting term and a single multiplier. That description makes it easier to check algebraic forms, track sign changes, and distinguish multiplicative patterns from additive ones.
Recognition is only the first step. The common ratio and starting term describe the terms, but additional reasoning is needed to answer questions about the value or convergence of an infinite series. Keep those questions separate: first identify the pattern accurately, then apply the appropriate results for the question being asked.
Check Your Understanding
For each question, focus on the relationship between consecutive terms. Where appropriate, state both the starting term and common ratio.
- Are \(10,-5,\frac52,-\frac54,\ldots\) geometric? Show the consecutive ratios.
- For \(a_n=7\left(\frac23\right)^{n-1}\), find \(a_1\), calculate \(a_2\) and \(a_3\), and identify the common ratio.
- Are \(1,4,16,64,\ldots\) geometric? What is the common ratio?
- Explain why \(3,7,11,15,\ldots\) is not geometric, even though the terms have a regular pattern.
- Does \(5,0,0,0,\ldots\) satisfy a geometric recurrence? If so, state the ratio and explain why a later quotient cannot be used.
