Question

evaluate the series.
\\(\displaystyle\sum_{i=1}^{6} (4i + 4)\\)
\\(\displaystyle\sum_{i=1}^{6} (4i + 4)=\boxed{?}\\)

Explanation:

Step1: Expand the summation

We know that $\sum_{i = 1}^{n}(a_{i}+b_{i})=\sum_{i = 1}^{n}a_{i}+\sum_{i = 1}^{n}b_{i}$. So, $\sum_{i = 1}^{6}(4i + 4)=4\sum_{i = 1}^{6}i+\sum_{i = 1}^{6}4$

Step2: Use summation formulas

The formula for $\sum_{i = 1}^{n}i=\frac{n(n + 1)}{2}$ and $\sum_{i = 1}^{n}c=cn$ (where $c$ is a constant). For $n = 6$, $\sum_{i = 1}^{6}i=\frac{6\times(6 + 1)}{2}=\frac{6\times7}{2}=21$ and $\sum_{i = 1}^{6}4=4\times6 = 24$

Step3: Calculate the result

Substitute the values back: $4\times21+24=84 + 24=108$

Answer:

108