Evaluating a finite sum using summation formulas

By linearity of summation, $\sum_{i=1}^{6}(2i+3) = 2\sum_{i=1}^{6} i + \sum_{i=1}^{6} 3$. Using the formula $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$ with $n=6$, we get $\sum_{i=1}^{6} i = \frac{6\cdot 7}{2} = 21$. Also, $\sum_{i=1}^{6} 3 = 6 \cdot 3 = 18$, since we are adding the constant 3 six times.

Substituting these values back in, $2\sum_{i=1}^{6} i + \sum_{i=1}^{6} 3 = 2(21) + 18 = 42 + 18 = 60$. So the sum evaluates to $60$.