Find a rule for a linear function $g$ such that $g\left(1\right) = 3.5$ and $g\left(4\right) = 5$.

1. Use the result from above to express $3.5 + 4 + 4.5 + 5$ in sigma notation.

   Step 1(a):

   Find the slope of the line going through $g\left(1\right)$ and $g\left(4\right)$.

   $\text{slope }=\frac{5-3.5}{4-1}=\frac{1.5}{3}=0.5$

   Step 2(a):

   Substitute the point and the slope in the point-slope form to find the equation.

   $\begin{array}{l} (g(x) - 5) = 0.5 (x - 4) \\ \qquad \; \, g(x) = 0.5x - 2 + 5 \\ \qquad \; \, g(x) = 0.5x + 3 \end{array}$

   Step 3(a):

   Write the Sigma notation.

   $\displaystyle\sum\limits_{k=1}^40.5k+3$

2. Find the sum in part (a).

   Hint (b):

   Add up the terms.