Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Find the sum of the first 10 terms of each arithmetic sequence. $$5,9,13, \dots$$

Step-by-Step Solution

Answer

The sum of the first 10 terms is 230.

Step 1: Identify the first term and common difference

The first term of the arithmetic sequence is given as \(a_1 = 5\). To find the common difference \(d\), subtract the first term from the second term: \(d = 9 - 5 = 4\).

Step 2: Use the formula for the sum of an arithmetic sequence

The formula for the sum of the first \(n\) terms of an arithmetic sequence is \(S_n = \frac{n}{2} (2a_1 + (n-1)d)\). Here, \(n = 10\), \(a_1 = 5\), and \(d = 4\).

Step 3: Substitute the values into the formula

Substitute the known values into the formula to find the sum of the first 10 terms: \(S_{10} = \frac{10}{2} (2 \times 5 + (10-1) \times 4)\). This simplifies to \(S_{10} = 5 (10 + 36)\).

Step 4: Calculate the sum

Simplify the expression to find the sum: \(S_{10} = 5 \times 46 = 230\). So, the sum of the first 10 terms is 230.

Key Concepts

Common Difference

In an arithmetic sequence, each term is generated by adding a constant value, known as the common difference, to the previous term. This constant difference between consecutive terms remains unchanged throughout the sequence. For instance, in the sequence you are working with: 5, 9, 13, ..., the common difference is determined by subtracting the first term from the second term. This is illustrated by the calculation:

• Common difference ( \(d\) ) = 9 - 5 = 4

Understanding the common difference is essential as it helps predict the sequence's behavior and is a fundamental component in other arithmetic calculations.

Sum of Terms

Understanding how to find the sum of the terms in an arithmetic sequence is extremely useful. It allows for quick calculation of larger sequences without manually adding each number. The formula used is:

• \(S_n = \frac{n}{2} (2a_1 + (n-1)d)\)

Where \(n\) is the number of terms, \(a_1\) is the first term, and \(d\) is the common difference.
For example, to find the sum of the first 10 terms in our sequence, substitute:

• \(n = 10\)
• \(a_1 = 5\)
• \(d = 4\)

Resulting expression: \(S_{10} = \frac{10}{2} (2 \times 5 + (10-1) \times 4)\), simplifying to \(S_{10} = 5 \times 46\), to find \(S_{10} = 230\).

Arithmetic Progression Formula

The arithmetic progression formula is a valuable tool in mathematics for analyzing sequences where each term is a constant difference from the previous one. It provides a straightforward method for determining the behavior and sum of the sequence. The basic formula to find any term in the sequence is:

• \(a_n = a_1 + (n-1) \cdot d\)

The afore mentioned formula can be tailored for more complex applications, like calculating a series sum.
By effectively applying this formula, one can understand and calculate the values within the sequence without listing each term individually. This efficiency is fundamental when dealing with extended series or when seeking a specific term. For example, finding the 10th term of the sequence:

• \(a_{10} = 5 + (10-1) \times 4 = 41\)

This formula bridges the gap between individual term calculation and broader sequence analysis.