What Is a Geometric Sequence?

A geometric sequence (or geometric progression, GP) is one where each term is obtained by multiplying the previous term by a constant called the common ratio (r).

Examples

2 6 18 54 162 ...

(r = 3, growth)

1000 500 250 125 ...

(r = 0.5, decay)

The nth Term Formula

T_n = ar^n-1

where a = first term, r = common ratio, n = term number

The common ratio is found by dividing any term by the previous term: r = T_n+1 / T_n.

Key property: If |r| > 1, the terms grow (diverge). If |r| < 1, the terms shrink (converge toward 0). If r is negative, terms alternate in sign.

Growth and Decay Applications

Compound Interest

$1000 at 5% pa: 1000, 1050, 1102.50, ... with r = 1.05

Population Growth

A colony doubling each year: 100, 200, 400, 800, ... with r = 2

Radioactive Decay

Half-life: 800, 400, 200, 100, ... with r = 0.5

Depreciation

A car losing 15% pa: value × 0.85 each year, r = 0.85

Key Vocabulary

Common Ratio (r)

The constant multiplier between consecutive terms: r = T_n+1 / T_n.

Geometric Mean

The geometric mean of two positive numbers a and b is √(ab). It is the middle term in a GP of three terms.

Growth Factor

When r > 1, the sequence exhibits exponential growth. The ratio r is the growth factor.

Decay Factor

When 0 < r < 1, the sequence decays toward zero. The ratio r is the decay factor.

Worked Examples

1

Find the 8th term of the GP: 3, 12, 48, 192, ...

Step 1: a = 3, r = 12/3 = 4.

Step 2: T₈ = 3 × 4⁷ = 3 × 16384 = 49152.

Answer: T₈ = 49,152.

2

A car worth $40,000 depreciates by 20% each year. Find its value after 5 years.

Step 1: a = 40000, r = 0.80 (retaining 80% each year).

Step 2: After 5 years: T₆ = 40000 × 0.80⁵ = 40000 × 0.32768 = 13107.20.

Answer: The car is worth $13,107.20 after 5 years.

3

The 3rd term of a GP is 12 and the 6th term is 96. Find a and r.

Step 1: T₃ = ar² = 12 and T₆ = ar⁵ = 96.

Step 2: Divide: ar⁵ / ar² = r³ = 96/12 = 8, so r = 2.

Step 3: a(4) = 12, so a = 3.

Answer: a = 3, r = 2. The sequence is 3, 6, 12, 24, 48, 96, ...

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

What is the common ratio of the sequence 5, 15, 45, 135, ...?

10 1/3 5 3

Question 2

Find the 6th term of the GP: 2, 10, 50, ...

6250 1250 31250 3125

Question 3

If a GP has a = 100 and r = 0.5, what happens to the terms as n increases?

They increase without bound They approach zero They stay constant They alternate in sign

Question 4

The geometric mean of 4 and 64 is:

34 32 16 256

Question 5

$10,000 is invested at 8% pa compound interest. What is the value after 3 years?

$12,400.00 $12,800.00 $12,000.00 $12,597.12

Geometric Sequences