Question

Eric and James are looking at this number sequence: 4, 8, 20, 56, 164. What must they do to find the term-to-term rule?

Answer 1

`f(n)=f(n-1) + 4(3)^(n-2)`

`\textsf{where}\quad f(1)=4`

Explanation:

Given sequence: 4, 8, 20, 56, 164

First, work out the difference in the terms:

`4 \underset{+4}{\longrightarrow} 8 \underset{+12}{\longrightarrow} 20 \underset{+36}{\longrightarrow} 56 \underset{+108}{\longrightarrow} 164`

As the first differences are not the same, work out the second differences:

`4 \underset{* 3}{\longrightarrow} 12 \underset{* 3}{\longrightarrow} 36 \underset{* 3}{\longrightarrow} 108`

Therefore, the term-to-term rule is:

`f(n)=f(n-1) + 4(3)^(n-2)`

`\textsf{where}\quad f(1)=4`

`f(2)=f(1) + 4(3)^(2-2)=4+ 4(3)^(0)=4+4=8`

`f(3)=f(2)+4(3)^(3-2)=8+4(3)^1=8+12=20`

`f(4)=f(3) + 4(3)^(4-2)=20+4(3)^2=20+36=56`

`f(5)=f(4) + 4(3)^(5-2)=56+4(3)^3=56+108=164`