Question

2) Write an explicit formula for the sequence.

\[\begin{cases}
c(1)=8 \\\\
c(n)=c(n-1)-13
\end{cases}\]
\[c(n)=\]

Answer 1

To find an explicit formula for the sequence defined by the recurrence relation:

\[
\begin{cases}
c(1) = 8 \\
c(n) = c(n-1) - 13
\end{cases}
\]

we notice that this is an arithmetic sequence where each term is obtained by subtracting 13 from the previous term.

The first few terms of the sequence can be calculated as follows:

- \(c(1) = 8\)
- \(c(2) = c(1) - 13 = 8 - 13 = -5\)
- \(c(3) = c(2) - 13 = -5 - 13 = -18\)
- \(c(4) = c(3) - 13 = -18 - 13 = -31\)

From this, we can see that the first term is \(8\) and the common difference \(d\) is \(-13\).

The general formula for the \(n\)-th term of an arithmetic sequence can be expressed as:

\[
c(n) = c(1) + (n - 1)d
\]

Substituting \(c(1) = 8\) and \(d = -13\):

\[
c(n) = 8 + (n - 1)(-13)
\]

Simplifying this further:

\[
c(n) = 8 - 13(n - 1)
\]
\[
= 8 - 13n + 13
\]
\[
= 21 - 13n
\]

Thus, the explicit formula for the sequence is:

\[
\boxed{c(n) = 21 - 13n}
\]