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

Nov 24, 2016

The next $4$ terms are:

$492 , 1464 , 4380 , 13128$

Recursive rule is:

$\left\{\begin{matrix}{a}_{1} = 8 \\ {a}_{n + 1} = 3 {a}_{n} - 12\end{matrix}\right.$

General formula is:

${a}_{n} = 2 \cdot {3}^{n - 1} + 6$

#### 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:

$\left\{\begin{matrix}{a}_{1} = 8 \\ {a}_{n + 1} = 3 {a}_{n} - 12\end{matrix}\right.$

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

${a}_{n} = A \cdot {3}^{n - 1} + B$

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

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

$\left\{\begin{matrix}A + B = 8 \\ 3 A + B = 12\end{matrix}\right.$

Subtracting the second equation from $3 \times$ the first equation, we find:

$2 B = 12$

Hence $B = 6$ and $A = 2$

So the general term of our sequence may be written:

${a}_{n} = 2 \cdot {3}^{n - 1} + 6$

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

${a}_{6} = 3 {a}_{5} - 12 = 3 \cdot 168 - 12 = 492$

${a}_{7} = 3 {a}_{6} - 12 = 3 \cdot 492 - 12 = 1464$

${a}_{8} = 3 {a}_{7} - 12 = 3 \cdot 1464 - 12 = 4380$

${a}_{9} = 3 {a}_{9} - 12 = 3 \cdot 4380 - 12 = 13128$