To find the explicit formula for the arithmetic sequence given, we first need to identify the first term and the common difference.
-
First term \( c(1) \): The first term is \( 10 \). So, \( c(1) = 10 \).
-
Common difference \( d \): To find the common difference, subtract the first term from the second term: \[ d = c(2) - c(1) = -10 - 10 = -20 \]
-
General formula for an arithmetic sequence: The formula for the \( n \)-th term of an arithmetic sequence can be expressed as: \[ c(n) = c(1) + (n - 1) \cdot d \] Substituting the values we found: \[ c(n) = 10 + (n - 1)(-20) \]
-
Simplifying the formula: \[ c(n) = 10 - 20(n - 1) \] \[ = 10 - 20n + 20 \] \[ = 30 - 20n \]
Thus, the explicit formula for the arithmetic sequence is: \[ c(n) = 30 - 20n \]