Geometric sequence (n-th term)

What the symbols mean

$a_n$

the n-th term

$a_1$

the first term (not 0)

$r$

the common ratio: what each term is multiplied by

$n$

the position of the term (1, 2, 3, …)

When to use it

Use it when each term is a fixed multiple of the one before: a ball bouncing to 60% of its height, a population doubling each generation, a value that loses 15% a year (r = 0.85). Solve for r from two known terms (an even power gives two possible ratios, ±), or for n to find when the sequence passes a value.

Why it works

Each step multiplies by r, and there are $n-1$ steps from the first term to the n-th, so the first term is multiplied by r a total of $n-1$ times.

Common mistakes

• Using $r^n$ instead of $r^{n-1}$, which gives the next term along (96 instead of 48 here).
• Writing a percentage decrease as the ratio: losing 15% a year means r = 0.85, not 0.15 or −0.15.
• Forgetting the negative root when solving for r from an even power: $r^4=81$ gives r = 3 or r = −3.