Question

Write an explicit formula for the sequence left-brace a subscript n baseline right-brace equals left-brace negative 3 comma negative Start Fraction 1 over 2 End Fraction comma 2 comma Start Fraction 9 over 2 End Fraction comma 7 comma ellipsis right-brace. Then find a subscript 17.
(1 point)
Responses

a subscript n baseline equals Start Fraction 5 over 2 End Fraction n minus Start Fraction 11 over 2 End Fraction semicolon 37
Image with alt text: a subscript n baseline equals Start Fraction 5 over 2 End Fraction n minus Start Fraction 11 over 2 End Fraction semicolon 37

a subscript n baseline equals Start Fraction 3 over 2 End Fraction n minus Start Fraction 9 over 2 End Fraction semicolon 21
Image with alt text: a subscript n baseline equals Start Fraction 3 over 2 End Fraction n minus Start Fraction 9 over 2 End Fraction semicolon 21

a subscript n baseline equals Start Fraction 5 over 2 End Fraction n minus Start Fraction 11 over 2 End Fraction semicolon Start Fraction 69 over 2 End Fraction
Image with alt text: a subscript n baseline equals Start Fraction 5 over 2 End Fraction n minus Start Fraction 11 over 2 End Fraction semicolon Start Fraction 69 over 2 End Fraction

a subscript n baseline equals negative Start Fraction 5 over 2 End Fraction n plus Start Fraction 11 over 2 End Fraction semicolon negative 37

Answer 1

To determine the explicit formula for the sequence \( \{ a_n \} = \{-3, -\frac{1}{2}, 2, \frac{9}{2}, 7, \ldots\} \), we first examine the pattern in the sequence values.

1. **Finding the First Differences:**
Let's calculate the differences between consecutive terms:
- \( -\frac{1}{2} - (-3) = 2.5 \)
- \( 2 - (-\frac{1}{2}) = 2.5 \)
- \( \frac{9}{2} - 2 = 2.5 \)
- \( 7 - \frac{9}{2} = 2.5 \)

The first differences are constant (all equal to \( 2.5 \)), indicating that the sequence is linear.

2. **Finding the General Form:**
A linear sequence can be expressed in the form:
\[
a_n = mn + b
\]
where \( m \) is the common difference and \( b \) is the first term of the sequence adjusted based on the index.

We already found that the common difference \( m = \frac{5}{2} \) because the difference is \( 2.5 \).

3. **Finding the Explicit Formula:**
We can find \( b \) using the first term:
- At \( n = 1 \), \( a_1 = -3 \):
\[
a_1 = m \cdot 1 + b = \frac{5}{2} \cdot 1 + b = -3 \implies b = -3 - \frac{5}{2} = -\frac{6}{2} - \frac{5}{2} = -\frac{11}{2}
\]

Now, our explicit formula for the sequence is:
\[
a_n = \frac{5}{2}n - \frac{11}{2}
\]

4. **Finding \( a_{17} \):**
Now, we calculate \( a_{17} \):
\[
a_{17} = \frac{5}{2} \cdot 17 - \frac{11}{2} = \frac{85}{2} - \frac{11}{2} = \frac{74}{2} = 37
\]

Therefore, the explicit formula is:
\[
a_n = \frac{5}{2}n - \frac{11}{2}
\]
and the value of \( a_{17} \) is:
\[
\boxed{37}
\]