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

Answer:

$f(n)=f(n-1) + 4(3)^{n-2}$

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

Step-by-step 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:

$4 \underset{\times 3}{\longrightarrow} 12 \underset{\times 3}{\longrightarrow} 36 \underset{\times 3}{\longrightarrow} 108$

Therefore, the term-to-term rule is:

$f(n)=f(n-1) + 4(3)^{n-2}$  

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

$f(2)=f(1) + 4(3)^{2-2}=4+ 4(3)^{0}=4+4=8$

$f(3)=f(2)+4(3)^{3-2}=8+4(3)^1=8+12=20$

$f(4)=f(3) + 4(3)^{4-2}=20+4(3)^2=20+36=56$

$f(5)=f(4) + 4(3)^{5-2}=56+4(3)^3=56+108=164$