Question

vvvvComplete the recursive formula of the arithmetic sequence
\[-5, -25, -45, -65,\unicode{0x2026}\].
\[b(1)=\]
\[b(n)=b(n-1)+\]

Answer 1

To complete the recursive formula of the arithmetic sequence given by the terms \(-5, -25, -45, -65, \ldots\), we first need to identify the consistent difference between the consecutive terms.

1. **Finding the common difference:**
- The difference between the first term \(-25\) and the second term \(-5\) is:
\[
-25 - (-5) = -25 + 5 = -20
\]

- The difference between the second term \(-45\) and the third term \(-25\) is:
\[
-45 - (-25) = -45 + 25 = -20
\]

- The difference between the third term \(-65\) and the fourth term \(-45\) is:
\[
-65 - (-45) = -65 + 45 = -20
\]

Thus, the common difference \(d\) is \(-20\).

2. **Setting up the recursive formula:**
- The first term of the sequence can be denoted as:
\[
b(1) = -5
\]

- The recursive formula can be written as:
\[
b(n) = b(n-1) + d
\]
Since \(d = -20\), we can substitute this value into the formula:
\[
b(n) = b(n-1) - 20
\]

Putting this together, we have:

\[
b(1) = -5
\]
\[
b(n) = b(n-1) - 20
\]

This is the complete recursive formula for the given arithmetic sequence.