Question

How do you write the next 4 terms in each pattern and write the pattern rule given 12, 36, 84, 180, 372?

Answer 1

The next `4` terms are: `756`, `1524`, `3060`, `6132`

A recursive rule is:

`{ (a_1 = 12), (a_(n+1) = 2a_n+12) :}`

A formula for the general term is:

`a_n = 3*2^(n+2)-12`

Explanation:

We can write a recursive rule for the sequence as follows:

`{ (a_1 = 12), (a_(n+1) = 2a_n+12) :}`

Dividing the sequence by `12` we get the sequence:

`1, 3, 7, 15, 31`

Compare this with the geometric sequence:

`2, 4, 8, 16, 32`

Notice that the terms of the original sequence divided by `12` are just `1` less than the terms of this sequence.

Hence we can write a general formula:

`a_n = (2^n-1)*12 = 3*2^(n+2)-12`

Use the recursive formula to find:

`a_6 = 2a_5+12 = 2*372+12 = 756`

`a_7 = 2a_6+12 = 2*756+12 = 1524`

`a_8 = 2a_7+12 = 2*1524+12 = 3060`

`a_9 = 2a_8+12 = 2*3060+12 = 60132`