How do you write the next 4 terms in each pattern and write the pattern rule given 7, 25, 79, 241, 727?

Nov 21, 2016

The next $4$ terms are: $2185$, $6559$, $19681$, $59047$

A recursive rule is:

$\left\{\begin{matrix}{a}_{1} = 7 \\ {a}_{n + 1} = 3 {a}_{n} + 4\end{matrix}\right.$

The general formula can be written:

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

Explanation:

A recursive formula for the given sequence is:

$\left\{\begin{matrix}{a}_{1} = 7 \\ {a}_{n + 1} = 3 {a}_{n} + 4\end{matrix}\right.$

That is: The first term is $7$ and each subsequent term is formed by multiplying by $3$ then adding $4$.

Consider the sequence of powers of $3$:

$1 , 3 , 9 , 27 , 81 , 243 , 729 , \ldots$

Notice that if we discard the first two terms and subtract $2$ from each of the resulting numbers then we get the given sequence.

So we can write a formula for a general term of the sequence as:

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

Using the recursive formula, we find:

${a}_{6} = 3 {a}_{5} + 4 = 3 \cdot 727 + 4 = 2185$

${a}_{7} = 3 {a}_{6} + 4 = 3 \cdot 2185 + 4 = 6559$

${a}_{8} = 3 {a}_{7} + 4 = 3 \cdot 6559 + 4 = 19681$

${a}_{9} = 3 {a}_{8} + 4 = 3 \cdot 19681 + 4 = 59047$