Evaluate the Summation sum from n=5 to 8 of 2x
The question is asking for a calculation of a mathematical series. Specifically, it involves performing the summation of the terms generated by the expression "2x" as x ranges from 5 to 8. This means that you would need to calculate the value of "2x" when x is 5, then add to it the value of "2x" when x is 6, continue by adding the value of "2x" when x is 7, and finally add the value of "2x" when x is 8. The question is essentially seeking the total of these four calculations.
$\sum_{n = 5}^{8} 2 x$
Answer
Solution:
Step 1: Write out the terms of the series
List each term of the summation for $n$ ranging from 5 to 8. The terms are $2x, 2x, 2x, 2x$.
Step 2: Combine the terms
Combine the terms of the series to find the sum.
Step 2.1: Combine the first two terms
Add together $2x$ and $2x$ to get $4x$.
Step 2.2: Add the next term
Add $4x$ to the next term $2x$ to get $6x$.
Step 2.3: Add the final term
Finally, add $6x$ to the last term $2x$ to get the total sum of $8x$.
The summation evaluates to $8x$.
Knowledge Notes:
The problem involves evaluating a finite arithmetic series. An arithmetic series is the sum of terms in an arithmetic sequence, where each term after the first is obtained by adding a constant difference to the previous term. In this case, the series is a constant series because each term is the same, namely $2x$.
To evaluate the summation of a constant series, you can simply multiply the constant term by the number of terms. Since the series is from $n=5$ to $n=8$, there are $8 - 5 + 1 = 4$ terms. Therefore, the sum can also be found by calculating $2x \times 4 = 8x$.
In the context of this problem, the summation symbol $\sum$ represents the operation of adding all terms of the series from the lower limit of summation ($n=5$) to the upper limit ($n=8$). The expression inside the summation, $2x$, is the general term of the series, which remains constant for each value of $n$ within the specified range.
The problem-solving process involves expanding the series, substituting the values, and simplifying the expression by performing the addition step by step until the final sum is obtained. This method is straightforward for a series with a small number of terms, as in this example.
