Question

) Write an explicit formula for the sequence.

\[\begin{cases}
b(1)=-22 \\\\
b(n)=b(n-1)+7
\end{cases}\]
\[b(n)=\]

Answer 1

The given sequence is defined recursively as follows:

\[
\begin{cases}
b(1) = -22 \\
b(n) = b(n-1) + 7
\end{cases}
\]

This indicates that the first term \(b(1)\) is \(-22\) and each subsequent term increases by \(7\).

To find an explicit formula for \(b(n)\), we can observe the pattern of the sequence:

- \(b(1) = -22\)
- \(b(2) = b(1) + 7 = -22 + 7 = -15\)
- \(b(3) = b(2) + 7 = -15 + 7 = -8\)
- \(b(4) = b(3) + 7 = -8 + 7 = -1\)
- \(b(5) = b(4) + 7 = -1 + 7 = 6\)

From this, we can see that each term increases by \(7\) for each increment in \(n\). The \(n\)-th term can be represented as:

\[
b(n) = b(1) + (n-1) \cdot 7
\]

Substituting \(b(1)\) into the formula:

\[
b(n) = -22 + (n-1) \cdot 7
\]

Now, simplifying this expression:

\[
b(n) = -22 + 7(n - 1)
\]

Distributing \(7\):

\[
b(n) = -22 + 7n - 7
\]
\[
b(n) = 7n - 29
\]

Thus, the explicit formula for the sequence is:

\[
\boxed{b(n) = 7n - 29}
\]