ACT Math: Use Geometric Sequences to Model Exponential Growth and Decay

Geometric sequence formula: aₙ=a₁·r⁽ⁿ⁻¹⁾, where a₁ is the first term, r is the common ratio, and n is the term number. Example: A bacteria population doubles every hour. Initial count: 100. After 5 hours: a₅=100·2⁽⁵⁻¹⁾=100·2⁴=100·16=1600. The sequence is 100, 200, 400, 800, 1600. Use this formula to find any term without calculating all previous ones.

To find the common ratio, divide any term by the previous term. If a sequence is 3, 6, 12, 24, then r=6/3=2. If a sequence is 100, 50, 25, 12.5, then r=50/100=0.5 (decay, r<1).

Error 1: Confusing the term number with the exponent. The exponent is (n-1), not n. So the 3rd term has exponent 2, not 3. Error 2: Misidentifying the common ratio. If a sequence decreases, r is a fraction, not a negative number. 100, 50, 25 has r=0.5, not r=-1. Always double-check your ratio by multiplying: Does a₁·r equal a₂?

Verification: Calculate the first three terms using your formula. If they match the given sequence, you've set up correctly.

Problem 1: A ball bounces to 80% of its previous height. Initial drop: 100 cm. After 4 bounces, what is the height? a₄=100·(0.8)³=100·0.512=51.2 cm. Problem 2: An investment triples every 10 years. Initial amount: $1,000. Value after 30 years (three 10-year periods)? a₃=1000·3²=9000. Problem 3: A radioactive substance decays by half every year. Start: 200 mg. After 5 years? a₅=200·(0.5)⁴=200·(1/16)=12.5 mg. Complete all three without a calculator; verify each answer makes sense.

Challenge: A sequence is 5, 15, 45, ... Find the 6th term. Answer: r=3, a₆=5·3⁵=5·243=1215.

ACT Math includes geometric sequences to test exponential thinking. Real-world applications (population growth, radioactive decay, compound interest) use geometric models. Expect 1-2 geometric sequence questions per Math section, often disguised as word problems about growth or decay.

Spend 20 minutes this week solving 6-8 geometric sequence problems. By test day, you'll identify the first term and common ratio in word problems instantly and use the formula to find any term without guessing.