Problem 63 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{i=1}^{5}(5 i+3)$$

Step-by-Step Solution

Answer

The sum of the series is 90.

Step 1: Understanding the Series

The problem asks us to find the sum of the series \( \sum_{i=1}^{5}(5i+3) \). This is an arithmetic series where each term is defined by the expression \(5i + 3\) for integer values of \(i\) from 1 to 5.

Step 2: Calculate Each Term

Calculate the terms of the series by plugging in values of \(i\) from 1 to 5:- For \(i=1\), term = \(5(1) + 3 = 8\)- For \(i=2\), term = \(5(2) + 3 = 13\)- For \(i=3\), term = \(5(3) + 3 = 18\)- For \(i=4\), term = \(5(4) + 3 = 23\)- For \(i=5\), term = \(5(5) + 3 = 28\)

Step 3: Summing the Terms

Now sum the terms that you calculated:\[ 8 + 13 + 18 + 23 + 28 \].

Step 4: Calculate the Final Sum

Add the numbers sequentially:- \(8 + 13 = 21\)- \(21 + 18 = 39\)- \(39 + 23 = 62\)- \(62 + 28 = 90\)Thus, the total sum is \( 90 \).

Key Concepts

Summation

Summation is a mathematical process of adding up a sequence of numbers to get a total sum. In arithmetic series, the summation notation \( \sum \) is often used. This notation signifies that elements of a sequence should be combined, or added together. For instance, in this problem, the symbol \( \sum_{i=1}^{5} \) tells us to add terms from \( i=1 \) to \( i=5\).When we write \( \sum_{i=1}^{5}(5i+3) \), it implies the sum of the series where each term is defined by the expression \( 5i + 3 \) for every \( i \) starting from 1 and going up to 5. The steps to find the sum are

• Calculate the value of each term by substituting different values of \( i \).
• Add small groups of terms together to verify intermediate results are correct.
• Finally, compute the total sum.

Understanding summation simplifies the process of dealing with long or complex series effortlessly.

Series Calculation

A series calculation involves evaluating a sum where each number in the sequence is found using a specific formula. In this exercise, it's an arithmetic series defined by the equation \( 5i + 3 \). Each number \( i \) in the series is plugged into this formula.

• The first step is to determine individual terms by changing \( i \) from 1 to 5. For example, for \( i = 1 \), \( 5(1) + 3 = 8 \); for \( i = 2 \), \( 5(2) + 3 = 13 \), and so on.
• Once all terms have been identified, they are summed together sequentially.
• It's helpful to double-check calculations for each term to avoid errors down the line.

This provides a methodical approach to accurately calculating the series sum.

Algebraic Expressions

Algebraic expressions form the backbone of arithmetic series and sums, as they allow us to generalize patterns and describe numbers succinctly. In this exercise, the arithmetic series is represented by the algebraic expression \( 5i + 3 \).

• An expression such as \( 5i + 3 \) denotes two operations: multiplication and addition. The variable \( i \) represents the term’s position in the sequence.
• When evaluating expressions for series, substitute consecutive integer values for \( i \) starting from 1 up to the number specified in the summation (in this case, 5).
• This general expression \( 5i + 3 \) helps to quickly find each term in the series without manual calculation every time.

Understanding algebraic expressions' mechanics can vastly improve efficiency and accuracy in handling sequences and series.