Geometric Sequences
Look for a constant multiplier—not a constant difference—then use the ratio to move forward, backward, or directly to any term.
Geometric Sequence
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A geometric sequence changes by the same multiplier each time
A sequence is geometric when each term is obtained by multiplying the previous term by one constant number, called the common ratio r.
Divide a term by the term before it.
aₙ = a₁rⁿ⁻¹
Divide by r.
Use 1 ± the decimal rate.
Example: In 5, 15, 45, 135, ... each term is multiplied by 3. Therefore r = 3. This differs from an arithmetic sequence, where the same amount is added or subtracted.
When a question asks for the total of the first n terms, use:
Sₙ = a₁(rⁿ − 1)(r − 1), r ≠ 1For 2, 6, 18, 54, 162, the sum of the first five terms is 242. You may add the terms directly when only a few are involved; the formula is faster for a longer list.
The exponent is n − 1
The first term has undergone zero multiplications, so a₁ = a₁r⁰.
A negative ratio alternates signs
Multiplying repeatedly by a negative number makes positive and negative terms alternate.
Divide to identify; multiply to continue
Example: Find the sixth term of 2, 6, 18, 54, ...
First confirm the ratio:
Now either continue multiplying or jump directly:
The exponent is 5 because moving from the first term to the sixth requires five multiplications.
Why it works
Checking two consecutive ratios prevents you from mistaking an irregular or arithmetic pattern for a geometric one.
Five forms you should recognize
Consider 160, 80, 40, 20, ...
80160 = 124080 = 12The constant ratio is 12, so every term is half the previous term.
If a₁ = 4 and r = 3, find a₅.
a₅ = a₁r⁵⁻¹a₅ = 4(3⁴) = 324From term 1 to term 5, the ratio is applied four times.
If a₄ = 54 and r = 3, then:
a₃ = 543 = 18a₂ = 183 = 6a₁ = 63 = 2Make 4, x, 100 geometric with a positive ratio.
x² = 4(100) = 400x = 20The middle term squared equals the product of its neighboring terms.
A quantity loses 20% each hour. It retains 80%, so its multiplier is 0.80.
250(0.80³) = 128After three hours, 128 units remain.
Check before you commit
- Using a constant difference instead of a constant ratio
- Writing rⁿ instead of rⁿ⁻¹ for the nth term
- Finding a ratio from only one pair and not checking the next pair
- Treating a 20% decrease as multiplication by 0.20 instead of 0.80
- Ignoring alternating signs when the ratio is negative
- Finding only the last term when a problem asks for the total
Do you need the lesson-or just practice?
One original question in each form recommends your next step. It does not yet verify mastery.
Work at the level you need.
Foundations
Build the core procedure with immediate explanations.
Core Practice
Use mixed forms with less scaffolding.
UPCAT-Style Transfer
Apply the competency in unfamiliar representations.
Ready to verify this competency?
A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.
Geometric Sequences FAQ
How is a geometric sequence different from an arithmetic sequence?
Arithmetic sequences add the same difference. Geometric sequences multiply by the same ratio.
Can the common ratio be zero or negative?
Yes. A negative ratio produces alternating signs. A zero ratio makes every term after the first equal to zero, though many school problems focus on nonzero ratios.
When do I use the sum formula?
Use a sum when the question asks for the total of several terms, not merely the amount in the final term.
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