Question

Find the next three terms of the sequence. Then write a rule for the sequence.

648, 216, 72, 24

Answer 1

By quick observation the difference between the terms is not constant, so the sequence isn't arithmetic. It does, however, appear to be geometric:$$r=\frac{216}{648}=\frac{72}{216}=\frac{24}{72}=\frac13$$... which means we can find the next three terms by multiplying by \(\frac13\):$$\frac13(24)=8;\frac13(8)=\frac83;\frac13\left(\frac83\right)=\frac89$$The explicit form of a geometric series is \(a_n=a_1r^{n-1}\); in our case, we've identified \(r=\frac13,\ a_1=658\) which yields \(a_n=658\left(\frac13\right)^{n-1}\)

Answer 2

648 **