Evaluate the following summation: 
${\sum }_{i=0}^2\frac{2i}{3}$

• A. $6$
• B. $\frac{2}{3}$
• C. $2$
• D. $\frac{4}{3}$

Answer: (C)

Verified step by step guidance

1. Identify the summation expression: \( \sum_{i=0}^2 \frac{2i}{3} \). This means you will evaluate the expression \( \frac{2i}{3} \) for each integer value of \( i \) from 0 to 2 and then sum the results.
2. Calculate the expression for \( i = 0 \): Substitute \( i = 0 \) into \( \frac{2i}{3} \) to get \( \frac{2 \times 0}{3} = 0 \).
3. Calculate the expression for \( i = 1 \): Substitute \( i = 1 \) into \( \frac{2i}{3} \) to get \( \frac{2 \times 1}{3} = \frac{2}{3} \).
4. Calculate the expression for \( i = 2 \): Substitute \( i = 2 \) into \( \frac{2i}{3} \) to get \( \frac{2 \times 2}{3} = \frac{4}{3} \).
5. Sum the results from each calculation: Add \( 0 + \frac{2}{3} + \frac{4}{3} \) to find the total sum of the series.