Problem 64 - A Graphical Approach to Precalculus with Limits

Question

Find the sum of each series. $$\sum_{i=1}^{5}(i-8)$$

Step-by-Step Solution

Answer

The sum of the series is -25.

Step 1: Understand the Series

We are given the series \( \sum_{i=1}^{5}(i-8) \). This means we need to substitute values of \( i \) from 1 to 5 into the expression \( i-8 \) and find the sum.

Step 2: Substitute Values into the Expression

Substitute \( i = 1, 2, 3, 4, \) and \( 5 \) into the expression \( i - 8 \).- For \( i = 1 \): \( 1 - 8 = -7 \)- For \( i = 2 \): \( 2 - 8 = -6 \)- For \( i = 3 \): \( 3 - 8 = -5 \)- For \( i = 4 \): \( 4 - 8 = -4 \)- For \( i = 5 \): \( 5 - 8 = -3 \)

Step 3: Calculate the Sum of the Series

Add up all the terms obtained from the substitution: \[ -7 + (-6) + (-5) + (-4) + (-3) \].Calculate the sum: \[ (-7) + (-6) = -13 \]\[ (-13) + (-5) = -18 \]\[ (-18) + (-4) = -22 \]\[ (-22) + (-3) = -25 \].Thus, the sum of the series is \(-25\).

Key Concepts

Summation

The process of summation involves adding a sequence of numbers to find their total. In mathematics, this is often represented using the sigma notation, which is a concise way to express large sums.
For example, the expression \( \sum_{i=1}^{5}(i-8) \) tells us to calculate the sum by substituting integers from 1 to 5 into the expression \( i-8 \) and then adding all the resulting values.

• The index \( i \) begins at 1 and runs through 5, based on the limits of the summation.
• Each term is evaluated individually, leading to separate values that are then added together.

Conducting the summation step-by-step as described, ensures that we don't make errors in our calculations.

Series Calculation

A series calculation refers to the addition of terms in a specific sequence. In our example, the series is derived from the difference expression \( i-8 \). After substituting values for \( i \), we compute each of the resulting terms and then sum these terms together to find the total.

• First, substitute the index values in the expression to get the individual terms. For instance, for \( i=3 \), find \( 3-8 = -5 \).
• Then, sequentially add all evaluated terms to find the sum: \( -7 + (-6) + (-5) + (-4) + (-3) = -25 \).

The process highlights careful substitution and arithmetic to reach the final result. Organizing each step helps in maintaining accuracy.

Precalculus

Precalculus combines algebra and trigonometry to prepare students for calculus. Within precalculus, understanding arithmetic series such as the one solved here is fundamental.
Such exercises solidify students’ abilities to evaluate expressions and sequences, which are foundational skills required for calculus and advanced mathematical problem solving.

• Recognizing patterns and noting how terms in a series change can aid in more advanced calculations.
• Understanding how to manipulate expressions and work with different kinds of sequences is crucial.

Arming yourself with these skills in precalculus sets the stage for tackling the derivative and integral concepts that are central to calculus.