What Is a Geometric Sequence? Formulas and Examples

A geometric sequence is a list of numbers in which each term is found by multiplying the previous term by the same constant value. This constant is called the common ratio. For example, 3, 6, 12, 24 follows a geometric pattern because every term is multiplied by 2.

These sequences appear in algebra, finance, science and everyday calculations. They can describe compound interest in an Australian savings account, repeated percentage changes in a business, or the way a quantity grows or decreases over time. Learning the pattern makes many longer calculations quicker and more reliable.

The key skills are recognising the common ratio, using the correct nth-term formula, and knowing when a sum is finite or infinite. A clear, step-by-step approach also helps students check whether an answer makes sense before entering it into a calculator.

Recognising The Common Ratio

To test whether a sequence is geometric, divide each term by the term immediately before it. If the result is always the same, the sequence has a common ratio, usually written as (r).

Consider 5, 15, 45, 135. Dividing gives (15 \div 5=3), (45 \div 15=3), and (135 \div 45=3). Therefore, (r=3). A sequence such as 4, 8, 12, 16 is not geometric because the quotients are different, even though the terms increase regularly.

The ratio can be negative or fractional. In 16, -8, 4, -2, the common ratio is (-\frac12), so the signs alternate while the absolute size of each term becomes smaller. In 80, 40, 20, 10, the ratio is (\frac12), showing decay rather than growth.

The Nth-Term Formula

The general formula for the (n)th term of a geometric sequence is

[ a_n=a_1r^{n-1} ]

Here, (a_n) is the term being found, (a_1) is the first term, (r) is the common ratio, and (n) is the position of the term. The exponent is (n-1) because the first term has been multiplied by the ratio zero times.

For the sequence 7, 21, 63, 189, the first term is 7 and the ratio is 3. The sixth term is

[ a_6=7(3)^{6-1}=7(3^5)=1701. ]

Writing the formula before substituting values reduces errors, especially when questions involve large term numbers or a ratio such as (\frac23).

Finding Missing Terms

A missing term can usually be found by identifying the ratio from the known values. If a sequence is 4, 12, __, 108, each term is multiplied by 3, so the missing value is 36. Checking (36\times3=108) confirms the result.

Sometimes the first term and a later term are given instead. Suppose (a_1=2) and (a_5=162). Applying the nth-term formula gives (162=2r^4), so (r^4=81). In a positive-ratio context, (r=3), producing the sequence 2, 6, 18, 54, 162.

When an equation produces more than one possible ratio, use the context and the sequence’s sign pattern. A negative ratio may be valid if the terms alternate, but it should not be selected automatically without checking every listed term.

Adding A Finite Sequence

The sum of the first (n) terms of a geometric sequence is

[ S_n=\frac{a_1(1-r^n)}{1-r}, \qquad r\ne1. ]

An equivalent version is

[ S_n=\frac{a_1(r^n-1)}{r-1}. ]

For 3, 6, 12, 24, the sum of the first four terms is

[ S_4=\frac{3(1-2^4)}{1-2} =\frac{3(-15)}{-1}=45. ]

This agrees with direct addition: (3+6+12+24=45). If the ratio is 1, every term is the same, so the sum is simply (S_n=na_1).

A practical example could involve a school fundraising plan in Brisbane where each weekly target is twice the previous target. The formula is faster than adding every week individually, particularly when the plan extends across a full term.

Infinite Geometric Series

An infinite geometric series continues without a final term. It has a finite sum only when the absolute value of the common ratio is less than 1. The formula is

[ S_\infty=\frac{a_1}{1-r}, \qquad |r|<1. ]

For (12+6+3+1.5+\cdots), (a_1=12) and (r=\frac12). Therefore,

[ S_\infty=\frac{12}{1-\frac12}=24. ]

The terms keep getting smaller, so the total approaches 24. If (r=2), the terms grow and no finite total exists. If (r=-1), the values alternate and do not settle towards one limit.

This idea can model repeated discounts or depreciation. For example, an item valued at $1,000 that loses 20% of its value each year follows a ratio of 0.8. The value after (n) years is (1000(0.8)^n), although the infinite sum of those values would describe a different question and should not be confused with the item’s resale value.

Applications In Everyday Maths

Compound interest is a common geometric-growth model. If an Australian bank account starts with $2,000 and earns 4% annually, the balance after (n) years is

[ A=2000(1.04)^n. ]

The percentage increase becomes the ratio: (1+0.04=1.04). For a 4% decrease, the ratio would be (1-0.04=0.96). This distinction is important when comparing savings products, term deposits or loan calculations in the local financial market.

Population estimates, repeated virus transmission models and business sales can also use geometric sequences when the percentage change stays consistent. A small online retailer serving customers in Sydney and Melbourne might forecast monthly sales by multiplying each month’s result by 1.1. Real data rarely follows a perfect ratio forever, but the model can still provide a useful first estimate.

For extra guided practice with mathematical problem-solving, students can also use online maths support alongside their class notes. Any external solution should still be checked by substituting the answer back into the original question.

Avoiding Common Mistakes

The most frequent error is confusing a common difference with a common ratio. In an arithmetic sequence, the same number is added or subtracted; in a geometric sequence, the same multiplier is used. The sequence 10, 15, 20, 25 is arithmetic, while 10, 15, 22.5, 33.75 is geometric with ratio 1.5.

Another mistake is using (n) instead of (n-1) in the nth-term formula. For (a_1=6) and (r=2), the first term must be (6(2)^0=6), not (6(2)^1=12). Calculator brackets also matter: enter (6\times2^{n-1}), rather than raising the entire expression to an unintended power.

Before finalising an answer, list the first few terms, inspect their signs and estimate their size. Students preparing for VCE, HSC or other Australian senior-secondary assessments can gain marks by showing the formula, substitution and simplification clearly. The method is often as important as the final number.

A geometric sequence is defined by repeated multiplication, and its common ratio controls whether the terms grow, shrink or alternate. Use (a_n=a_1r^{n-1}) for an individual term, the finite-sum formula for a fixed number of terms, and (S_\infty=\frac{a_1}{1-r}) only when (|r|<1). The essential reminder is simple: identify the ratio first, then choose the formula that matches the question.