Geometric Sequence Term Calculator

How it is calculated

Each step multiplies by the common ratio, so reaching position n takes (n − 1) multiplications of the first term.

aₙ = a₁ × r^(n − 1)

a₁ is the first term, r the common ratio (a plain factor, so 1.05 means +5 % per step and 0.5 halves each time) and n the position counted from 1. The position must be a whole number of 1 or more; 0, negative and fractional values give no result. A negative ratio flips the sign at every step. Ordinary answers are shown to 4 decimals, but a term of 1,000,000,000 or more, or a non-zero term below 0.01 in size, is shown in scientific notation with six significant figures (for example 1.90735 × 10⁻⁶). If the term is too large for the number range (3 to the power 1998, say), there is no result.

Common mistakes

• Entering 5 instead of 1.05 for a 5 % rise. The ratio is the growth factor, not the percentage, so +5 % is 1.05 and −5 % is 0.95.
• Using n as the number of steps. Position n has n − 1 steps behind it, so after 10 periods you need n = 11.
• Forgetting that a negative ratio alternates the signs of the terms (2, −6, 18, −54, …), so position 4 with first term 2 and ratio −3 is −54.
• Expecting a result for a fractional or zero position. Terms exist only at whole positions from 1.
• Reading a result like 1.90735 × 10⁻⁶ as an error. It is 0.00000190735; very small or very large terms switch to this notation.