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

Nov 21, 2016

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

A recursive rule is:

$\left\{\begin{matrix}{a}_{1} = 12 \\ {a}_{n + 1} = 2 {a}_{n} + 12\end{matrix}\right.$

A formula for the general term is:

${a}_{n} = 3 \cdot {2}^{n + 2} - 12$

#### Explanation:

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

$\left\{\begin{matrix}{a}_{1} = 12 \\ {a}_{n + 1} = 2 {a}_{n} + 12\end{matrix}\right.$

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:

${a}_{n} = \left({2}^{n} - 1\right) \cdot 12 = 3 \cdot {2}^{n + 2} - 12$

Use the recursive formula to find:

${a}_{6} = 2 {a}_{5} + 12 = 2 \cdot 372 + 12 = 756$

${a}_{7} = 2 {a}_{6} + 12 = 2 \cdot 756 + 12 = 1524$

${a}_{8} = 2 {a}_{7} + 12 = 2 \cdot 1524 + 12 = 3060$

${a}_{9} = 2 {a}_{8} + 12 = 2 \cdot 3060 + 12 = 60132$