Evaluate the Summation sum from x=1 to 4 of 2x
The question asks to calculate the total sum of a mathematical series where the term being added each time is twice the variable \(x\). The summation range provided is between \(x=1\) and \(x=4\), meaning that you need to find the sum of the series by plugging in the values of \(x\) from 1 to 4 into the formula \(2x\) and then adding all the resulting values together.
$\sum_{x = 1}^{4} 2 x$
Answer
Solution:
Step 1:
Write out the terms of the summation for each value of $x$: $2 \cdot 1 + 2 \cdot 2 + 2 \cdot 3 + 2 \cdot 4$
Step 2:
Perform the simplification.
Step 2.1:
Calculate $2$ times $1$: $2 + 2 \cdot 2 + 2 \cdot 3 + 2 \cdot 4$
Step 2.2:
Calculate $2$ times $2$: $2 + 4 + 2 \cdot 3 + 2 \cdot 4$
Step 2.3:
Combine $2$ and $4$: $6 + 2 \cdot 3 + 2 \cdot 4$
Step 2.4:
Calculate $2$ times $3$: $6 + 6 + 2 \cdot 4$
Step 2.5:
Combine $6$ and $6$: $12 + 2 \cdot 4$
Step 2.6:
Calculate $2$ times $4$: $12 + 8$
Step 2.7:
Combine $12$ and $8$: $20$
Knowledge Notes:
The problem involves evaluating a finite arithmetic series, which is a sequence of numbers with a common difference between consecutive terms. In this case, the series is $2x$ for $x$ ranging from $1$ to $4$. The process of evaluating the series involves the following steps:
Expansion of the Series: The first step is to write out each term of the series explicitly. This is done by substituting the values of $x$ into the expression $2x$.
Simplification: The next step is to simplify the expanded series by performing the arithmetic operations. This includes multiplication of the constant factor (in this case, $2$) with each value of $x$, followed by the addition of the resulting products.
Arithmetic Operations: Each step of the simplification process involves basic arithmetic operations: multiplication and addition.
Final Summation: The last step is to add up all the simplified terms to get the final sum of the series.
In mathematical notation, the summation of a series is often represented using the sigma notation $\sum$, which compactly represents the sum of a sequence of terms. The problem given does not require the use of any advanced formulas for arithmetic series since it's a simple case where each term can be calculated and added manually. However, for longer series, one might use the formula for the sum of an arithmetic series, which is given by $S_n = \frac{n}{2}(a_1 + a_n)$, where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, and $a_n$ is the nth term.
