Evaluate the Summation sum from k=5 to 13 of (5-2k)/8

Solution:

Step 1:

Adjust the summation's starting index to 1 by separating the original summation into two parts.

$$\sum_{k = 5}^{13} \frac{5 - 2k}{8} = \sum_{k = 1}^{13} \frac{5 - 2k}{8} - \sum_{k = 1}^{4} \frac{5 - 2k}{8}$$

Step 2:

Calculate the first summation $\sum_{k = 1}^{13} \frac{5 - 2k}{8}$.

Step 2.1:

Break down the summation into two simpler summations.

$$\sum_{k = 1}^{13} \frac{5}{8} - \sum_{k = 1}^{13} \frac{2k}{8}$$

Step 2.2:

Compute the summation of the constant term $\sum_{k = 1}^{13} \frac{5}{8}$.

Step 2.2.1:

Apply the constant summation formula.

$$\sum_{k = 1}^{n} c = cn$$

Step 2.2.2:

Insert the given values into the formula.

$$\frac{5}{8} \cdot 13$$

Step 2.2.3:

Perform the multiplication.

$$\frac{5 \cdot 13}{8}$$ $$\frac{65}{8}$$

Step 2.3:

Compute the summation involving $k$, $\sum_{k = 1}^{13} \frac{2k}{8}$.

Step 2.3.1:

Extract the constant factor from the summation.

$$-\frac{1}{4} \sum_{k = 1}^{13} k$$

Step 2.3.2:

Utilize the arithmetic series formula.

$$\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$$

Step 2.3.3:

Substitute the values into the formula and include the constant factor.

$$-\frac{1}{4} \cdot \frac{13(13 + 1)}{2}$$

Step 2.3.4:

Simplify the expression.

$$-\frac{1}{4} \cdot \frac{13 \cdot 14}{2}$$ $$-\frac{1}{4} \cdot \frac{182}{2}$$ $$-\frac{91}{4}$$

Step 2.4:

Combine the results of the two summations.

$$\frac{65}{8} - \frac{91}{4}$$

Step 2.5:

Simplify the expression to a common denominator.

Step 2.5.1:

Convert $-\frac{91}{4}$ to have a denominator of 8.

$$\frac{65}{8} - \frac{91 \cdot 2}{8}$$

Step 2.5.2:

Combine the numerators over the common denominator.

$$\frac{65 - 182}{8}$$

Step 2.5.3:

Simplify the numerator.

$$\frac{-117}{8}$$

Step 3:

Calculate the second summation $\sum_{k = 1}^{4} \frac{5 - 2k}{8}$.

Step 3.1:

Write out the terms for each value of $k$.

$$\frac{5}{8} - \frac{1}{4} + \frac{5}{8} - \frac{1}{2} + \frac{5}{8} - \frac{3}{4} + \frac{5}{8} - 1$$

Step 3.2:

Simplify the series.

Step 3.2.1:

Combine like terms.

$$\frac{20}{8} - \frac{10}{4}$$

Step 3.2.2:

Reduce the fractions to simplest form.

$$\frac{5}{2} - \frac{5}{2}$$

Step 3.2.3:

Subtract the fractions.

$$0$$

Step 4:

Substitute the summation results with their evaluated values.

$$- \frac{117}{8} - 0$$

Step 5:

Combine the final terms.

$$- \frac{117}{8}$$

Step 6:

Present the result in various formats.

Exact Form:

$$- \frac{117}{8}$$

Decimal Form:

$$- 14.625$$

Mixed Number Form:

$$- 14 \frac{5}{8}$$

Knowledge Notes:

The problem involves evaluating a finite summation, which is a common task in algebra and calculus. The relevant knowledge points include:

1. Summation Notation: The sigma notation $\sum$ is used to represent the sum of a sequence of terms. The index of summation and the upper and lower bounds indicate the range of values to be summed.

2. 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 a specific case of an arithmetic series, where the difference between consecutive terms is constant.

3. Summation of a Constant: The sum of a constant $c$ over $n$ terms is simply $cn$.

4. Simplification of Fractions: When adding or subtracting fractions, a common denominator is required. Multiplying by an appropriate form of 1 (e.g., $\frac{2}{2}$) can help achieve a common denominator without changing the value of the fractions.

5. Algebraic Manipulation: The problem requires algebraic manipulation, including distributing constants through a summation, combining like terms, and simplifying expressions.

6. Multiple Representations of Numbers: The final answer can be presented in different forms, such as an exact fraction, a decimal, or a mixed number, depending on the context or preference.