Question

Identify the correct explicit formula for the following sequence.

7, 2, -3, -8, . . .

(1 point)
Responses

f(n)=7+(n−1)(−5)f of n is equal to 7 plus open paren n minus 1 close paren times negative 5

f(n)=7(−15)n−1f of n is equal to 7 times open paren negative 1 fifth close paren raised to the n minus 1 power

f(n)=7−(n−1)15f of n is equal to 7 minus open paren n minus 1 close paren times 1 fifth

f(n)=7(−5)n−1

Answer 1

To find the explicit formula for the given sequence \(7, 2, -3, -8, \ldots\), we need to examine the pattern of the sequence.

1. **Identify the First Term**:
The first term \( f(1) = 7 \).

2. **Identify the Common Differences**:
The differences between consecutive terms are:
- \(2 - 7 = -5\)
- \(-3 - 2 = -5\)
- \(-8 - (-3) = -5\)

The common difference is \(-5\), indicating that this is an arithmetic sequence.

3. **General Formula for Arithmetic Sequence**:
The general formula for the \(n\)-th term of an arithmetic sequence is:
\[
f(n) = f(1) + (n-1) \cdot d
\]
where \(d\) is the common difference.

4. **Substituting Values**:
- \(f(1) = 7\)
- \(d = -5\)

Substituting these values into the formula gives us:
\[
f(n) = 7 + (n-1)(-5)
\]

Thus, the correct explicit formula for the sequence is:

**Response**: \( f(n) = 7 + (n-1)(-5) \)