Advanced · 15 min
Arithmetic and geometric sequences
Find a term from its position in two common types of sequence.
Use a constant difference
An arithmetic sequence adds the same difference d each time. If its first term is a, term n is a + (n − 1)d, because the first term needs no step.
Worked example
For 4, 7, 10, …, term 5 is 4 + 4 × 3 = 16.
Use a constant ratio
A geometric sequence multiplies by the same ratio r each time. Term n is a × rⁿ⁻¹. The exponent counts the multiplications after the first term.
Worked example
For 3, 6, 12, …, term 5 is 3 × 2⁴ = 48.
Write a rule
- Write the nth-term rule for 7, 12, 17, … .
- Use the rule to find term 20 without listing all earlier terms.