\( -n^{2}+a n+b \Rightarrow \) is 5 and The finst of the first difference is 7 . Calulate the value of \( a \) and \( b \)

To find the values of \( a \) and \( b \), we start with the expression \( -n^2 + an + b \). Given that the function takes a value of 5 when \( n \) is at its first term (which we can assume is 0), we have the equation: When \( n = 0 \): \[ b = 5 \] Now, let's calculate the first difference. The first difference is defined as the change in the function's value as \( n \) increases by 1. Thus, we need to evaluate the function at \( n = 1 \): When \( n = 1 \): \[ f(1) = -1^2 + a \cdot 1 + b = -1 + a + 5 = a + 4 \] The first difference when moving from \( n = 0 \) to \( n = 1 \) is: \[ \Delta f(0) = f(1) - f(0) = (a + 4) - 5 = a - 1 \] Since we know this first difference equals 7, we can set up the equation: \[ a - 1 = 7 \] \[ a = 8 \] Now we have \( a = 8 \) and \( b = 5 \). Thus, the values are: \[ a = 8, \quad b = 5 \]