Geometric Sequences, Common Ratios, and nth-Term Rules

A geometric sequence changes by multiplying by the same number at each step. That fixed multiplier is the common ratio, r. In 3, 6, 12, 24, …, divide each term by the previous one: 6/3 = 2, 12/6 = 2, and 24/12 = 2. The common ratio is 2, so each term doubles.

Find the common ratio

Divide a term by the term immediately before it: r = a₂/a₁, or generally aₙ/aₙ₋₁. Keep the order consistent. In 80, 40, 20, 10, …, the common ratio is 40/80 = 1/2. The values shrink by half each step. A negative ratio alternates signs: 2, −6, 18, −54 has ratio −3.

If the ratio changes, the sequence is not geometric. For example, 2, 5, 8, 11 has a constant difference of 3, so it is arithmetic. Compare both the differences and the ratios when a sequence's pattern is not obvious.

Use the explicit nth-term formula

The direct formula is aₙ = a₁rⁿ⁻¹. The exponent n − 1 counts how many times the first term is multiplied to reach term n. The first term uses zero multiplications, so a₁r⁰ = a₁. For 3, 6, 12, 24, …, aₙ = 3(2)ⁿ⁻¹. The seventh term is 3(2)⁶ = 192.

Verify the formula at n = 1: 3(2)⁰ = 3. At n = 2, 3(2)¹ = 6. These checks catch the common error of using exponent n, which shifts every term one place.

Write a recursive rule

A recursive description gives the starting value and says how each next value is formed. For 3, 6, 12, …, state a₁ = 3 and aₙ = 2aₙ₋₁ for n ≥ 2. This is convenient for generating consecutive terms. The explicit formula is faster when the problem asks for a distant term.

Model growth and decay

A ratio greater than 1 in absolute value makes magnitudes grow; a positive ratio between 0 and 1 makes terms approach zero. For instance, an amount that increases 5% per period is multiplied by 1.05 each period. If it begins at 200 units, after n periods it is 200(1.05)ⁿ. An amount that decreases by 5% per period is multiplied by 0.95, not by −0.05.

The percentage change is not the multiplier itself. A 20% increase means multiply by 1 + 0.20 = 1.20; a 20% decrease means multiply by 1 − 0.20 = 0.80. Identify whether the sequence describes a fixed additive change or a repeated percent change before choosing an arithmetic or geometric model.

If an account begins at $500 and grows by 4% once per year, the yearly balances follow 500, 520, 540.80, … . The common ratio is 1.04, so after five annual growth periods the amount is 500(1.04)⁵, approximately $608.33 before fees or taxes. The increase is not a fixed $20 each year: the 4% is applied to the new balance each time, so later dollar increases become larger.

An exponential sequence has a constant ratio when each step is one equal time period. By contrast, linear growth has a constant difference. In the first year, the account increases by $20; in the next year, it increases by $20.80. Those changing differences but stable 1.04 multiplier indicate geometric growth. If a question gives a table, compare both consecutive differences and quotients to identify the pattern.

Find a missing position or ratio

If you know two terms, their position numbers matter. In a geometric sequence, aₙ/aₖ = rⁿ⁻ᵏ when the ratio is nonzero. If a₁ = 5 and a₄ = 135, then 135/5 = r³, so r³ = 27 and r = 3. The sequence begins 5, 15, 45, 135. For an even power, such as r² = 9, both r = 3 and r = −3 may be possibilities; check all information in the sequence.

Common errors

• Subtracting terms when the pattern uses multiplication.
• Dividing in the wrong order when finding the common ratio.
• Using exponent n rather than n − 1 in the nth-term rule.
• Treating a 10% increase as a multiplier of 0.10 instead of 1.10.
• Assuming a negative ratio means every term is negative; signs alternate when the first term is positive.
• Ignoring that ratios require a nonzero previous term for division.

A final check is to divide two consecutive terms predicted by your rule. Their ratio should equal r. Also confirm that the rule gives the stated first term. If a word problem uses a repeated percent change, calculate the number of periods carefully: 'after five years' normally means five multiplications after the starting amount.

For a decreasing quantity, use the same structure. If 80% of a material remains after each treatment, r = 0.80. Starting with 200 grams, the amount after four treatments is 200(0.80)⁴ = 81.92 grams. The model predicts a fraction of the previous amount each time, not a fixed subtraction of 20% of the original 200 grams. A fixed subtraction of 40 grams would be arithmetic and would reach zero after five steps; repeated 20% decay approaches zero without following that linear path.

Write an explicit rule from the first term and ratio

If the first term is a₁ and common ratio is r, the nth term is aₙ = a₁rⁿ⁻¹. The exponent is n − 1 because the first term has undergone zero multiplications. For a sequence beginning 5, 15, 45, the ratio is 3 and aₙ = 5·3ⁿ⁻¹. The fourth term is 5·3³ = 135.

A negative common ratio alternates signs: 2, −6, 18, −54 has r = −3. A ratio between 0 and 1 produces terms that decrease in magnitude when the first term is positive; a ratio less than −1 alternates and grows in magnitude. Interpret the sign and magnitude separately.

Exam takeaway

Check ratios of consecutive terms to recognize geometric change. Use aₙ = a₁rⁿ⁻¹, where n − 1 is the number of multiplications after the first term. In growth and decay problems, convert percent changes into multipliers and count the periods from the initial value.