Question

Write an explicit formula for the sequence left-brace a subscript n baseline right-brace equals left-brace 7 comma 2 comma negative 3 comma negative 8 comma negative 13 comma ellipsis right-brace. Then find a subscript 14 baseline.
(1 point)
Responses

a subscript n baseline equals negative 5 n plus 2 semicolon negative 58
Image with alt text: a subscript n baseline equals negative 5 n plus 2 semicolon negative 58

a subscript n baseline equals negative 5 n plus 2 semicolon negative 68
Image with alt text: a subscript n baseline equals negative 5 n plus 2 semicolon negative 68

a subscript n baseline equals negative 5 n plus 12 semicolon negative 68
Image with alt text: a subscript n baseline equals negative 5 n plus 12 semicolon negative 68

a subscript n baseline equals negative 5 n plus 12 semicolon negative 58
Image with alt text: a subscript n baseline equals negative 5 n plus 12 semicolon negative 58

Answer 1

To determine the explicit formula for the given sequence \( \{ a_n \} = \{ 7, 2, -3, -8, -13, \ldots \} \), we first need to observe the pattern in the sequence.

Calculating the differences between consecutive terms:

- \( 2 - 7 = -5 \)
- \( -3 - 2 = -5 \)
- \( -8 - (-3) = -5 \)
- \( -13 - (-8) = -5 \)

The sequence decreases consistently by 5, which indicates it is an arithmetic sequence with:

- First term \( a_1 = 7 \)
- Common difference \( d = -5 \)

The formula for the nth term of an arithmetic sequence is given by:

\[
a_n = a_1 + (n-1)d
\]

Substituting the known values:

\[
a_n = 7 + (n-1)(-5)
\]

This simplifies to:

\[
a_n = 7 - 5(n-1)
\]
\[
a_n = 7 - 5n + 5
\]
\[
a_n = 12 - 5n
\]

Now, to find \( a_{14} \):

\[
a_{14} = 12 - 5(14)
\]
\[
a_{14} = 12 - 70
\]
\[
a_{14} = -58
\]

Thus, the explicit formula is:

\[
a_n = 12 - 5n
\]

And the answer for \( a_{14} \) is \( -58 \).

The correct response is:

**a subscript n baseline equals negative 5 n plus 12; negative 58**