Geometric Progression (G.P.)
Simple Explanation
A geometric progression (G.P.) is a sequence where each term is found by multiplying the previous term by the same fixed number, the common ratio r. Its nth term is given directly by aₙ = a₁·rⁿ⁻¹.
Why Do We Need It?
G.P.s model any process that changes by the same fixed percentage or factor each step — compound interest, population growth, and radioactive decay all follow this pattern.
Formula
nth Term of a Geometric Progression
aₙ = a₁ · rⁿ⁻¹
Gives the value of any term of a geometric sequence directly from its position n.
- aₙ
- — the nth term of the sequence
- a₁
- — the first term
- r
- — the common ratio — the fixed factor multiplied between consecutive terms
- n
- — the term's position in the sequence
When to use it: Whenever you need a specific term of a geometric sequence directly, or need to find n, a₁, or r from other known values.
Worked Example
Find a term of a geometric progression
A G.P. has first term 3 and common ratio 2. Find the 6th term.
Why Does This Work?
Starting at a₁ and multiplying by r exactly (n−1) times reaches the nth term, for the same reason the A.P. formula works — reaching aₙ from a₁ always takes exactly n−1 "steps," here each step being a multiplication by r instead of an addition of d.
Real-Life Example
Compound interest growth
An investment of $1000 grows by 5% each year, so its value each year forms a G.P. with a₁=1000, r=1.05.
The nth term formula gives the exact investment value after any number of years directly.
Practice
A G.P. has first term 5 and common ratio 3. Find the 4th term.
MediumA G.P. has a₁=2 and a₄=54. Find the common ratio r.
MediumCommon mistake
Using n instead of (n−1) as the exponent of r — the first term a₁ requires zero multiplications by r, so its exponent is (1−1)=0, giving r⁰=1.
Quick Review
- aₙ = a₁ · rⁿ⁻¹.
- r = common ratio: the fixed factor multiplied between consecutive terms.
- Models any process changing by a fixed percentage/factor each step.