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