The first six numbers in a pattern are listed below. \[ -19.2-18.4 \quad-17.6 \quad-16.8 \quad-16 \quad-15.2 \] Write an expression that could be used to determine the 100 th number in the pattern. You must show work.

To find an expression for the \( n \)-th number in the given pattern, we first observe the numbers: \[ -19.2, -18.4, -17.6, -16.8, -16, -15.2 \] We can see that the pattern increases by \( 0.8 \) with each subsequent number. Let's denote the first term as \( a_1 \) and the common difference as \( d \). We have: - \( a_1 = -19.2 \) - \( d = -18.4 - (-19.2) = 0.8 \) The formula for the \( n \)-th term of an arithmetic sequence is given by: \[ a_n = a_1 + (n - 1) \cdot d \] Substituting the values we found into the formula gives: \[ a_n = -19.2 + (n - 1) \cdot 0.8 \] To find the \( 100 \)-th term, substitute \( n = 100 \): \[ a_{100} = -19.2 + (100 - 1) \cdot 0.8 \] \[ a_{100} = -19.2 + 99 \cdot 0.8 \] \[ a_{100} = -19.2 + 79.2 \] \[ a_{100} = 60 \] Thus, the expression to determine the \( 100 \)-th number in the pattern is: \[ a_n = -19.2 + (n - 1) \cdot 0.8 \] And the 100-th number is \( 60 \).