Statistics & Probability
Geometric Sequences
A geometric sequence has a constant ratio (r) between terms. The nth term formula is: aₙ = a₁ × r^(n−1).
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Worked Examples
Follow along with these step-by-step examples. Take your time — there's no rush!
Problem
Find the 6th term of: 2, 6, 18, 54, ...
Step-by-Step Solution
Common ratio r = 6/2 = 3. First term a₁ = 2.
aₙ = a₁ × r^(n−1)
a₆ = 2 × 3^(6−1) = 2 × 3⁵ = 2 × 243 = 486
Answer
486
Problem
Find the common ratio and next two terms: 80, 40, 20, 10, ...
Step-by-Step Solution
r = 40/80 = 1/2
Next: 10 × 1/2 = 5, then 5 × 1/2 = 2.5
Answer
r = 1/2; next terms are 5 and 2.5
Problem
Write a formula for: 3, 12, 48, 192, ...
Step-by-Step Solution
a₁ = 3, r = 12/3 = 4.
aₙ = 3 × 4^(n−1)
Answer
aₙ = 3 × 4^(n−1)
Problem
Is 1/8 a term in the sequence 8, 4, 2, 1, ...?
Step-by-Step Solution
r = 4/8 = 1/2. Formula: aₙ = 8 × (1/2)^(n−1).
Set 8 × (1/2)^(n−1) = 1/8.
(1/2)^(n−1) = 1/64 = (1/2)⁶ → n−1 = 6 → n = 7.
Yes, 1/8 is the 7th term.
Answer
Yes, it is the 7th term
Problem
A bacteria population doubles every hour. Start: 50. How many after 5 hours?
Step-by-Step Solution
This is a geometric sequence with a₁ = 50 and r = 2.
a₆ = 50 × 2^(6−1) = 50 × 32 = 1600.
Answer
1600 bacteria after 5 hours
Your Turn — Practice Problems
Try all 5 problems on your own first. Write out your work — that's how it sticks!
💡 Tip: Don't peek at the answers until you've genuinely tried each one.
Find the 5th term: 1, 3, 9, 27, ...
What is the common ratio: 100, 50, 25, 12.5, ...?
Find the 4th term: 5, 15, 45, ...
Write nth term formula: 4, 8, 16, 32, ...
Find the 7th term: 2, 6, 18, 54, ...
What is the common ratio: 3, 12, 48, 192, ...?
Find the 6th term: 1, 2, 4, 8, ...
Write nth term formula: 1, 5, 25, 125, ...
Find the 3rd term: 100, 10, 1, ...
A population of 200 triples each year. How many after 3 years?
Is 64 a term in: 1, 2, 4, 8, ...?
Find the 4th term: 1000, 100, 10, ...
Finished all 12? Give yourself a pat on the back — then check your work!
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