Evaluate the Summation sum from x=0 to 2 of 2x+2
The given problem asks for the calculation of a summation where the term to be summed is given by the formula 2x+2. Specifically, it requests that you perform this summation for all integer values of x starting at 0 and ending at 2. The problem essentially wants you to find the total of 2x+2 for x=0, x=1, and x=2, and then add those values together to get the final result.
$\sum_{x = 0}^{2} 2 x + 2$
Answer
Solution:
Step 1:
Write out the terms of the summation for each value of \( x \) within the given range.
\( 2 \cdot 0 + 2 \) for \( x = 0 \) \( 2 \cdot 1 + 2 \) for \( x = 1 \) \( 2 \cdot 2 + 2 \) for \( x = 2 \)
Step 2:
Perform the calculations.
Step 2.1:
Calculate \( 2 \cdot 0 \).
\( 0 + 2 \) for \( x = 0 \)
Step 2.2:
Combine the result of \( 2 \cdot 0 \) with \( 2 \).
\( 2 \) for \( x = 0 \)
Step 2.3:
Calculate \( 2 \cdot 1 \).
\( 2 + 2 \) for \( x = 1 \)
Step 2.4:
Combine the result of \( 2 \cdot 1 \) with \( 2 \).
\( 4 \) for \( x = 1 \)
Step 2.5:
Sum the results for \( x = 0 \) and \( x = 1 \).
\( 2 + 4 \)
Step 2.6:
Calculate \( 2 \cdot 2 \).
\( 4 + 2 \) for \( x = 2 \)
Step 2.7:
Combine the result of \( 2 \cdot 2 \) with \( 2 \).
\( 6 \) for \( x = 2 \)
Step 2.8:
Sum the results for \( x = 0 \), \( x = 1 \), and \( x = 2 \).
\( 2 + 4 + 6 \)
Step 2.9:
Combine all sums to get the final result.
\( 12 \)
Knowledge Notes:
The problem requires evaluating a finite summation, which is a process of adding up all the values of a function at different points within a certain range. The function in this case is \( f(x) = 2x + 2 \), and the range is from \( x = 0 \) to \( x = 2 \).
To solve this, we follow these steps:
Expansion: We expand the summation by calculating the function's value at each integer within the given range.
Simplification: We simplify the expression by performing the indicated operations, such as multiplication and addition.
Combination: We combine the simplified terms to find the sum total.
The summation notation \( \sum \) is a concise way to represent the addition of a sequence of numbers. The expression \( \sum_{x=a}^{b} (2x + 2) \) means "sum the expression \( 2x + 2 \) for all values of \( x \) from \( a \) to \( b \)".
In this problem, the summation is finite and only involves three terms, making it straightforward to evaluate by hand. The process involves basic arithmetic operations: multiplication and addition.
