Sequences and Series

Geometric Sequences

Find the nth term of a geometric sequence using a_n = a_1 · r^(n-1), where r is the common ratio.

Example

Use the geometric sequence formula to find a term.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

a1=2r=3n=5a_{ 1 }= 2 \quad r= 3 \quad n= 5

Step 2 — Formula

Use the geometric sequence formula.

an=a1rn−1a_n=a_{ 1 }r^{n- 1 }

Step 3 — Substitute

Substitute first term 2, ratio 3, and exponent 4.

a5=2⋅34a_{ 5 }= \hl{2} \cdot \hl{3} ^{ \hl{4} }

Step 4 — Result

The fifth term is 162.

a5=162a_{ 5 }= \hl{162}
geometric-sequence A geometric sequence has a constant ratio r between consecutive terms: a_{n+1} / a_n = r for all n. The nth term formula: a_n = a_1 · r^(n-1). - a_1 is the first term - r is the common ratio - n-1 is the exponent (power)