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
Doubles, Triples, etc word problems are the same as geometric sequence
• Total = Initial (increase)ᵗᶦᵐᵉ (increase can be in terms such as doubles, triples, etc)
EXAMPLES
• The number of toys a child owns triples every 2 years. If he has 100 toys at age 1. How many toys does one have at age 9?
T = 100 (3)⁸
PRACTICE PROBLEMS
Question 1 What is the 5th term in the geometric sequence, -4, 8, -16, 32…?
-- Answer -- -64
sequence: -4, 8, -16, 32, -64, 128
5th term: -64
Question 2 Given the 1st term in a sequence is 5. The terms in the sequence have a constant ratio of 5. What is the 4th term?
-- Answer -- 625
2nd term: 5*5 = 25
3rd term: 25*5 = 125
4th term: 125*5 = 625
Question 3 Houses in Milwaukee triple every year. If there are 25 houses in a neighborhood of Milwaukee now, how many houses will be in that neighborhood at the end of 5 years?
-- Answer -- 6075
Total = initial * (3)ᵗᶦᵐᵉ = 25 (3)⁵ = 25 * 243 = 6075
ACT PRACTICE PROBLEMS