Evaluate the Summation sum from k=1 to 40 of -2k+8
The question asks for the evaluation of a specific summation expression. The summation involves adding together the terms generated by the formula "-2k + 8" for each integer value of k starting at k=1 and ending at k=40. Essentially, you are being asked to calculate the total sum after applying this formula to each k in the specified range.
$\sum_{k = 1}^{40} - 2 k + 8$
Answer
Solution:
Step:1
Decompose the given summation into two separate summations using the properties of summation. $$\sum_{k = 1}^{40} (-2k + 8) = -2 \sum_{k = 1}^{40} k + \sum_{k = 1}^{40} 8$$
Step:2
Calculate $-2 \sum_{k = 1}^{40} k$.
Step:2.1
Use the arithmetic series summation formula:
$$\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$$
Step:2.2
Insert the upper limit of the summation into the formula and include the coefficient:
$$-2 \left( \frac{40(40 + 1)}{2} \right)$$
Step:2.3
Proceed with the simplification.
Step:2.3.1
First, perform the addition inside the parentheses.
$$-2 \frac{40 \cdot 41}{2}$$
Step:2.3.2
Next, simplify the multiplication within the fraction.
$$-2 \left( \frac{1640}{2} \right)$$
Step:2.3.3
Reduce the fraction by removing the common factor of 2.
$$-1 \cdot 1640$$
Step:2.3.4
Finally, multiply the remaining terms.
$$-1640$$
Step:3
Calculate $\sum_{k = 1}^{40} 8$.
Step:3.1
Apply the formula for the summation of a constant term:
$$\sum_{k = 1}^{n} c = cn$$
Step:3.2
Place the constant and the upper limit into the formula.
$$(8)(40)$$
Step:3.3
Execute the multiplication.
$$320$$
Step:4
Combine the results from the two summations.
$$-1640 + 320$$
Step:5
Add the two values together to get the final result.
$$-1320$$
Knowledge Notes:
Summation Properties: Summation (denoted by the sigma symbol $\sum$) has properties that allow us to break down complex summations into simpler parts. For instance, the summation of a sum of terms can be separated into the sum of individual summations.
Arithmetic Series: The sum of the first $n$ natural numbers is given by the formula $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$. This is an arithmetic series with a common difference of 1.
Summation of a Constant: The sum of a constant $c$ over $n$ terms is simply $cn$, because the constant does not change as the index increases.
Simplification: When simplifying expressions, it's important to perform operations in the correct order, cancel common factors, and combine like terms to reach the simplest form.
Combining Results: After evaluating separate parts of a summation, the final step often involves adding or subtracting the results to find the total sum.
