Finding the Common Ratio

$r = \frac{a_2}{a_1} = \frac{a_3}{a_2} = ...$

Divide any term by the previous term.

Explicit Formula

To find the $n$th term: $a_n = a_1 \cdot r^{n-1}$

where:

• $a_n$ = nth term
• $a_1$ = first term
• $r$ = common ratio
• $n$ = term number

Recursive Formula

$a_n = a_{n-1} \cdot r$

Each term equals the previous term times $r$.

Growth vs. Decay

Growth: $|r| > 1$

• Terms get larger in magnitude

Decay: $0 < |r| < 1$

• Terms get smaller

Real-World Applications

• Compound interest
• Population growth
• Radioactive decay

$r = \frac{15}{5} = 3$

Check: $\frac{45}{15} = 3$ ✓

Answer: Common ratio = $3$