Geometric Sequences
Use common ratios to model repeated multiplication.
A geometric sequence is formed by multiplying by the same number each time. This number is the common ratio, written \(r\).
Nth term formula\[u_n=u_1r^{n-1}\]
Worked example
For \(5, 15, 45, 135,\ldots\), \(u_1=5\) and \(r=3\).
\[u_n=5\cdot 3^{n-1}\]The 8th term is \(u_8=5\cdot3^7=10935\).
Connection to modelling
Repeated percentage change is geometric. Increasing by 6% repeatedly means multiplying by \(1.06\) each time.