Question

How do you write the next 4 terms in each pattern and write the pattern rule given 8, 12, 24, 60, 168?

Answer 1

The next `4` terms are:

`492, 1464, 4380, 13128`

Recursive rule is:

`{ (a_1 = 8), (a_(n+1) = 3a_n-12) :}`

General formula is:

`a_n = 2*3^(n-1)+6`

Explanation:

Given:

`8, 12, 24, 60, 168`

Notice that all of the terms are divisible by `4`, so consider the sequence formed by dividing by `4`:

`2, 3, 6, 15, 42`

Write down the sequence of differences between consecutive terms:

`1, 3, 9, 27`

Notice that these are just powers of `3`.

Hence we can deduce a recursive rule for the original sequence:

`{ (a_1 = 8), (a_(n+1) = 3a_n - 12) :}`

This sequence must also have a general formula of the form:

`a_n = A*3^(n-1) + B`

where `A, B` are constants to be determined.

Putting `n = 1, 2` to get two equations to solve, we find:

`{ (A+B = 8), (3A+B = 12) :}`

Subtracting the second equation from `3 xx` the first equation, we find:

`2B = 12`

Hence `B=6` and `A=2`

So the general term of our sequence may be written:

`a_n = 2*3^(n-1) + 6`

Using the recusive formulation, we can find the next `4` terms:

`a_6 = 3a_5-12 = 3*168-12 = 492`

`a_7 = 3a_6-12 = 3*492-12 = 1464`

`a_8 = 3a_7-12 = 3*1464-12 = 4380`

`a_9 = 3a_9-12 = 3*4380-12 = 13128`