To find an expression for the sequence given, let's first identify the pattern. The sequence is:
-74, -148, -222, -296, ...
We can find the common difference between consecutive terms:
-148 - (-74) = -74
-222 - (-148) = -74
-296 - (-222) = -74
The common difference \( d \) is -74.
This is an arithmetic sequence where:
- The first term \( a_1 = -74 \)
- The common difference \( d = -74 \)
The general formula for the \( n \)-th term of an arithmetic sequence is given by:
\[ a_n = a_1 + (n - 1) \cdot d \]
Substituting \( a_1 \) and \( d \):
\[ a_n = -74 + (n - 1) \cdot (-74) \]
Simplifying the expression:
\[
a_n = -74 - 74(n - 1)
\]
\[
a_n = -74 - 74n + 74
\]
\[
a_n = -74n
\]
Now, we can find the 27th term by substituting \( n = 27 \):
\[ a_{27} = -74 \cdot 27 \]
Calculating:
\[ a_{27} = -1998 \]
Thus, the final answers are:
\[ a_n = -74n \]
\[ a_{27} = -1998 \]
1 answer