Question

What is the recursive form uo for this sequence?

14, 18, 22, 26, 30,…

Answer 1

To find the recursive form for the given sequence \(14, 18, 22, 26, 30, \ldots\), we can first determine the pattern in the sequence.

1. The first term \(u_1\) is \(14\).
2. The second term \(u_2\) is \(18\).
3. The third term \(u_3\) is \(22\).
4. The fourth term \(u_4\) is \(26\).
5. The fifth term \(u_5\) is \(30\).

We can see that the difference between consecutive terms is constant:

\[
u_2 - u_1 = 18 - 14 = 4
\]
\[
u_3 - u_2 = 22 - 18 = 4
\]
\[
u_4 - u_3 = 26 - 22 = 4
\]
\[
u_5 - u_4 = 30 - 26 = 4
\]

This shows that the sequence is an arithmetic sequence with a common difference of \(4\).

We can establish a recursive formula using the first term and the common difference. The recursive form can be written as:

\[
u_n = u_{n-1} + 4 \quad \text{for } n \geq 2
\]

And the initial condition is:

\[
u_1 = 14
\]

Putting it all together, the recursive definition for the sequence is:

\[
\begin{align*}
u_1 & = 14 \\
u_n & = u_{n-1} + 4 \quad \text{for } n \geq 2
\end{align*}
\]