Problem 45 - College Algebra

Question

For Exercises \(45-50,\) write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. $$\sum_{i=1}^{17}(5 i+3)$$

Step-by-Step Solution

Answer

The first three terms are 8, 13, and 18. The last term is 88. The sum of the first 17 terms is 816.

Step 1: Calculate the First Terms

Find the first three terms of the sequence by replacing \(i\) with 1, 2, and 3 in the equation \(5i +3\). This operation results in the first term being 8, the second term being 13, and the third term being 18.

Step 2: Calculate the Last Term

Find the last term of the sequence by replacing \(i\) with 17 in the equation \(5i +3\). This will give the last term as 88.

Step 3: Using the Sum Formula

The sum of the first 17 terms of the arithmetic sequence can be calculated using the formula \(S_n = \frac{n}{2}(a + l)\). Here, \(n\) is 17 (the number of terms), \(a\) is 8 (the first term), and \(l\) is 88 (the last term). Substituting these values into the formula, we obtain \(S_{17} = \frac{17}{2}(8 + 88)\), so the sum will be 816.

Key Concepts

Arithmetic Sequence

An arithmetic sequence, also known as an arithmetic progression, is a series of numbers in which each term after the first is obtained by adding a constant, called the common difference, to the previous term. For example, in the sequence 2, 4, 6, 8,..., the common difference is 2. This is a foundational concept in algebra, where sequences are ordered lists of numbers following a particular pattern.

Understanding how to identify the common difference and the subsequent terms of an arithmetic sequence is crucial. In our exercise, the sequence is determined by the algebraic expression \(5i + 3\). To find the first three terms, we simply substitute \(i\) with 1, 2, and 3, reflecting the sequence's iterative nature as it progresses by adding the constant \(5\) to each new term.

Series and Summation

A series is the sum of the terms of a sequence. When dealing with an arithmetic sequence, the series is the sum of all the arithmetic terms from the first to the last. Summation is the process of calculating the total sum of these terms. The capital Greek letter sigma \( \Sigma \) is commonly used to represent the sum of a sequence, and it is followed by an expression that provides the terms to be added up.

In arithmetic sequences, summation becomes a repetitive addition where each term increases by the common difference. Our textbook problem \( \sum_{i=1}^{17}(5i + 3) \) involves finding the sum of the first 17 terms of the sequence, where \(i\) ranges from 1 to 17. Summation in arithmetic series is streamlined by a specific formula, making the calculation more efficient than adding each term individually.

Algebraic Expressions

Algebraic expressions are combinations of letters and numbers where the letters represent variables. These expressions express operations to be performed on the variables, often including addition, subtraction, multiplication, and division. In the context of sequences and series, algebraic expressions define the terms of the sequence.

The expression \(5i + 3\) in our example consists of the variable \(i\), representing the position of each term in the sequence, and the constants 5 and 3. Algebraic expressions enable us to describe each term's value in relation to its position, providing a direct way to calculate any term in the sequence without listing all preceding terms.

Arithmetic Series Formula

The arithmetic series formula is a shortcut to find the sum of an arithmetic sequence without adding each term separately. It leverages the structure of the sequence and only requires the first term, the last term, and the total number of terms. The formula is given by \( S_n = \frac{n}{2}(a + l) \), where \( S_n \) is the sum of the first \( n \) terms, \( a \) is the first term, \( l \) is the last term, and \( n \) is the total number of terms.

Applying this formula to our example, with the first term of 8, the last term of 88, and 17 terms in total, we find the sum of the series to be 816 easily. This formula significantly simplifies the process and is a powerful tool for quickly solving summation problems in arithmetic sequences.