Problem 37 - Algebra for College Students

Question

Find the sum of the first 40 terms of the arithmetic sequence \(2,6,10,14,18, \ldots\) 3200

Step-by-Step Solution

Answer

The sum of the first 40 terms is 3200.

Step 1: Identify the first term and common difference

First, we need to identify the first term of the sequence and the common difference. The first term is given as \(a_1 = 2\). To find the common difference \(d\), we subtract the first term from the second term: \(d = 6 - 2 = 4\).

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

The sum of the first \(n\) terms of an arithmetic sequence can be found using the formula: \[ S_n = \frac{n}{2} \cdot (2a + (n - 1)d) \] where \(S_n\) is the sum of the first \(n\) terms, \(a\) is the first term, \(n\) is the number of terms, and \(d\) is the common difference.

Step 3: Plug values into the formula

We need to find the sum of the first 40 terms, so \(n = 40\). We already have \(a = 2\) and \(d = 4\). Substituting these into the formula gives: \[ S_{40} = \frac{40}{2} \cdot (2 \times 2 + (40 - 1) \times 4) \].

Step 4: Simplify the expression

Calculating inside the parentheses first: \(2 \times 2 = 4\) and \((40 - 1) \times 4 = 156\). Therefore, inside the parentheses we have: \(4 + 156 = 160\).

Step 5: Calculate the sum

Substitute back into the expression: \[ S_{40} = 20 \times 160 \]. Now, calculate the product: \(20 \times 160 = 3200\).

Step 6: Conclusion

The sum of the first 40 terms of the arithmetic sequence is \(3200\).

Key Concepts

Sum of Arithmetic Sequence

Understanding the sum of an arithmetic sequence is crucial when you deal with patterns that change steadily. An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. The total addition of the numbers in a sequence up to a certain number of terms is called the sum of the sequence.
To find this sum, we use the formula:

• \[ S_n = \frac{n}{2} \cdot (2a + (n - 1)d) \]

• \(S_n\) denotes the sum of the first \(n\) terms.
• \(n\) represents the number of terms you are summing.
• \(a\) and \(d\) stand for the first term and the common difference, respectively.

Exploring sequences like \(2, 6, 10, \ldots\) and wanting to find the sum of the first 40 terms, the sequence formula simplifies your work by offering a precise way to calculate this.

First Term

The first term in an arithmetic sequence initiates the pattern of the sequence. It is usually denoted by \(a_1\). In the problem we are discussing, the first term is easily seen as 2. It's the starting point of the series and a key variable in the sequence formula. Identifying the first term correctly is essential, as it forms the foundation for calculating subsequent terms using the common difference.
Knowing the first term:

• Guides you to understand what value the sequence starts from.
• Helps in applying the arithmetic sequence formulas accurately.
• Is vital in solving for the sequence's sum.

Without the right first term, calculations for the rest of the sequence would be off-target. Therefore, ensure that you have identified \(a_1\) accurately by observing the sequence closely.

Common Difference

The common difference, denoted as \(d\), is a constant amount that you add to each term to get to the next one in an arithmetic sequence. This difference remains unchanged throughout the sequence.
To find \(d\), simply subtract the first term from the second term. In our scenario:

• \[ d = 6 - 2 = 4 \]

The common difference:

• Defines the interval between each term in the series.
• Ensures the sequence's linear progression.
• Influences the value of the nth term and the sum of a set number of terms.

By maintaining consistency with this difference, you guarantee that the series follows the mathematical rule for arithmetic sequences.

Sequence Formula

The sequence formula makes calculating various elements of an arithmetic series much easier. This magical tool allows you to determine both individual terms and the total sum of a sequence efficiently. It generally takes the form:

• \[ a_n = a_1 + (n - 1)\times d \]

Where:

• \(a_n\) symbolizes the nth term.
• \(a_1\) is the first term, pivotal in determining where the sequence begins.
• \(d\) is the common difference, ensuring regular increments between terms.
• \(n\) determines the position of the term you want to find.

By using these formulas, anyone can unravel complex sequential patterns with ease. The sequence formula is not just a tool but acts as a bridge connecting basic concepts like the first term, common difference, and the sequence's progression towards finding sums or specific terms.