What is a Geometric Sequence?. Learn what makes a sequence geometric: a constant common ratio r found by dividing consecutive terms. Spot geometric patterns, calculate r for any sequence, and see how the sign and magnitude of r affect how the terms grow, shrink, or alternate.
The General Term Formula. Build the formula t_n = t_1 times r to the power n minus 1 from first principles using a pattern table. Apply it to find specific terms, write simplified general terms, and follow order-of-operations rules to avoid calculator errors.
Working with the Formula. Master every question type: listing terms from a given general term formula, counting terms in a finite geometric sequence, finding missing terms and geometric means, and solving for x when three algebraic expressions must form a geometric sequence.
Applications. Apply geometric sequences to real-world growth and decay problems including car depreciation and bacteria doubling. Learn to identify the correct common ratio for percent-change situations and carefully count n from the starting term.
Inside the lesson — a free preview
Definition
Geometric Sequence. A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant value. This constant is called the common ratio, written as r.
Arithmetic sequence — add d each time
Geometric sequence — multiply by r each time
Worked example: Spotting a Geometric Pattern
To spot a geometric sequence, ask: what do I multiply each term by to get the next one?
Example — The sequence -3, 9, -27, 81, . Step 1 — Divide consecutive terms:
9 ÷ (-3) = -3
-27 ÷ 9 = -3
81 ÷ (-27) = -3
Step 2 — Identify r:
Each term is multiplied by -3 to get the next. So r = -3 and this is a geometric sequence.
Step 3 — Continue the pattern:
81 × (-3) = -243
-243 × (-3) = 729
729 × (-3) = -2187
-2187 × (-3) = 6561
Next 4 terms: -243, 729, -2187, 6561
Key Observation — Negative r. When r is negative, the terms alternate between positive and negative. You can spot this pattern immediately without doing any calculations.
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