Geometric sequences
CONCEPTS
Geometric Sequence
• A series of numbers that changes by the same rate (or there is a common ratio)
• Look for words, "geometric sequence" or "constant ratio" in the question
Geometric Sequence: series of numbers that change by the same rate (r)
• r = aₙ / aₙ₋₁
• aₙ = a₁ * r ⁿ⁻¹
EXAMPLES
Given the sequence, what is the ratio?
2,4,6,8,10 => ratio = 2
3,12,48,192 => ratio = 4
PRACTICE PROBLEMS
Question 1 What is the 5th term in the geometric sequence, –10, 5…?
-- Answer -- –5/8
d = a₂ / a₁ = 5 / –10 = –1/2
d = a₃ / a₂ = –1/2 = a3 / 5 => a₃ = –5/2
sequence: –10, 5, –5/2, 5/4, –5/8…
5th term: –5/8
Question 2 What is the 5th term if the 4th term is 64 and the 2nd term is 4?
-- Answer -- 256
Method 1
a₄ = a₂ * r², 64 = 4 * r² => r = 4
a₅ = r a₄ = 4 * 64 = 256
Method 2
Term Number Ratio
1
2 4
3 4r
4 64 4r²
5 4r³
4r² = 64, r = 4
4 (4)³ = 256
ACT PRACTICE PROBLEMS