Question

\begin{cases}d(1)=3\\\\
d(n)=d(n-1)-14
\end{cases} Find the
\[3^{\text{rd}}\] term in the sequence.

Answer 1

To solve for the \(n\)-th term of the sequence defined by the recurrence relation, we start with the base case and use it to find additional terms.

The initial condition is given as:
\[
d(1) = 3.
\]

The recurrence relation states:
\[
d(n) = d(n-1) - 14.
\]

Now we will compute the terms step by step.

1. **Finding \(d(2)\)**:
\[
d(2) = d(1) - 14 = 3 - 14 = -11.
\]

2. **Finding \(d(3)\)**:
\[
d(3) = d(2) - 14 = -11 - 14 = -25.
\]

Now we have calculated \(d(1)\), \(d(2)\), and \(d(3)\):
- \(d(1) = 3\)
- \(d(2) = -11\)
- \(d(3) = -25\)

Thus, the third term in the sequence is:
\[
\boxed{-25}.
\]