nth Term of a Geometric Progression: Formula & Examples

The nth term of a geometric progression is the formula that lands directly on any term without multiplying through all the earlier ones. If the first term is a and the common ratio is r, the nth term is T_n = a times r^(n - 1). The exponent is (n - 1) because travelling from the 1st term to the nth term applies the multiplier exactly (n - 1) times.

This formula matters whenever repeated multiplication runs for many steps. Counting bacteria after 30 generations, finding the value of an investment after 25 years of compounding, or determining the 12th note in a musical scale all need one term far down the line. The nth-term formula delivers it with a single power calculation.

Key Concepts

• n: the position number, a positive whole number.
• a: the first term of the GP.
• r: the common ratio, the same multiplier between every pair of consecutive terms.
• Exponent (n - 1): T_1 = a times r^0 = a, T_2 = a times r^1, T_3 = a times r^2. The exponent always trails the term number by one.
• Sign behaviour: a negative r makes the terms alternate in sign; a ratio with absolute value below 1 makes the terms shrink toward zero.

Formulas

• nth term: T_n = a times r^(n - 1).
• Linking two terms: T_n = T_m times r^(n - m), handy when the first term is unknown but a later term is given.
• Recovering r: r = T_(n+1) / T_n, using any two consecutive terms.
• Assumptions: n is a positive integer and r stays constant across the sequence.

Worked Example

Find the 8th term of the GP with first term a = 5 and common ratio r = -2. Then T_8 = 5 times (-2)^7. Since (-2)^7 = -128, T_8 = 5 times (-128) = -640. Listing the terms confirms the sign pattern: 5, -10, 20, -40, 80, -160, 320, -640. Every even-numbered term is negative because an odd power of a negative ratio stays negative, and the 8th term -640 matches the formula.

Common Misconceptions

• T_n = a times r^n: the familiar off-by-one slip. Checking n = 1 exposes it at once: a times r^1 would wrongly multiply the first term by r.
• Negative ratios are impossible: they are completely valid and simply make the sequence alternate signs, as in 5, -10, 20, -40, … . Many real processes, like alternating-current samples, behave this way.
• r must be rational: r can be any non-zero real number, including irrationals. A GP with r = square root of 2, such as 1, 1.414, 2, 2.828, …, is legitimate, and such ratios appear in paper sizes and musical scales.

Real-World Applications

• Microbiology: the cell count after n generations of binary fission is a times 2^(n - 1).
• Finance: the balance after n years at interest rate i is P times (1 + i)^(n - 1) measured from the first year-end.
• Music: each semitone multiplies frequency by the 12th root of 2, so the nth note follows a GP.
• Discounts: successive discounts of 20% then 10% multiply the price by 0.8 then 0.9, a two-step GP.

The nth term of a GP, T_n = a times r^(n - 1), compresses (n - 1) multiplications into one power. Watch the exponent carefully, respect the sign when r is negative, and remember the formula also runs sideways: any term can be reached from any other term through T_n = T_m times r^(n - m).