Problem 56 - Algebra for College Students

Question

Find the sum of the first 40 terms of the arithmetic sequence with the general term \(a_{n}=4 n-7 . \quad 3000\)

Step-by-Step Solution

Answer

The sum of the first 40 terms is 3000.

Step 1: Understand the Formula for an Arithmetic Sequence

The general term of the arithmetic sequence is given by \(a_{n} = 4n - 7\). This means the first term, \(a_1\), is found by substituting \(n=1\) into the formula. We also need to identify the number of terms \(n = 40\) and the formula for the nth term of an arithmetic sequence is \(a_n = a_1 + (n-1) \cdot d\), where \(d\) is the common difference.

Step 2: Find the First Term and Common Difference

To find the first term \(a_1\), we substitute \(n=1\) into the formula: \(4(1) - 7 = -3\). The general term formula indicates the difference between consecutive terms. Therefore, the common difference, \(d\), is the coefficient of \(n\), which is 4.

Step 3: Use the Sum Formula for Arithmetic Series

To find the sum of the first 40 terms \(S_{40}\), we use the formula for the sum of an arithmetic sequence: \[ S_n = \frac{n}{2} \times (a_1 + a_n) \]. We need \(a_n\), the last term, found by substituting \(n=40\) into \(a_n = 4n - 7\).

Step 4: Calculate the Last Term

Calculate \(a_{40}\): Substitute \(n=40\) in \(4(40) - 7 = 160 - 7 = 153\). So, \(a_{40} = 153\).

Step 5: Calculate the Sum of the Series

Now substitute the known values into the sum formula: \[ S_{40} = \frac{40}{2} \times (-3 + 153) \]. Compute the operations: \(S_{40} = 20 \times 150 = 3000\).

Key Concepts

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. Imagine you have a sequence where every number is generated by repeatedly adding a fixed number to the previous one. The sum of such numbers is what we call an arithmetic series.
This sequence creates a harmonious pattern, making it easy to identify how each term is related to its predecessor and successor.

• In our example, we're working with a sequence defined by the general term formula \(a_n = 4n - 7\).
• To find the sum of these terms, we use an arithmetic series formula, which simplifies the process considerably.

Sum of Arithmetic Sequence

Finding the sum of an arithmetic sequence involves a handy formula. You will often use:\[S_n = \frac{n}{2} \times (a_1 + a_n)\]where:

• \(S_n\) is the sum of the first \(n\) terms,
• \(a_1\) is the first term of the sequence,
• \(a_n\) is the nth term.

In our exercise, since we need the sum of the first 40 terms, we'd use this formula with our first term \(a_1 = -3\) and the 40th term \(a_n = 153\).
By plugging these into the formula \(S_{40} = \frac{40}{2} \times (-3 + 153)\), we get the sum of 3000.

Common Difference

The common difference in an arithmetic sequence is the constant value added to each term to produce the next. It directly influences the specific values in your sequence.
In a formula, it is denoted as \(d\). We determine this by looking at the general term equation or by subtracting any term from its successor.

• For example, in the sequence from our exercise, the general term was \(a_n = 4n - 7\).
• The coefficient of \(n\) in this equation is 4, which means our common difference \(d = 4\).

The common difference is what makes the sequence arithmetic, ensuring that each term follows the pattern evenly.

General Term Formula

The general term formula allows us to discover any term in an arithmetic sequence quickly. It's a straightforward way to describe each term without having to write out the entire sequence. The formula is \(a_n = a_1 + (n-1) \cdot d\).

• Here, \(a_n\) is the nth term of the sequence,
• \(a_1\) is the first term, and
• \(d\) is the common difference.

In our given sequence, the formula provided is already a generalized expression: \(a_n= 4n - 7\).
This formula not only helps in verifying terms but also is essential when calculating sums or exploring the sequence's properties.
So if you wanted to find out, say, the tenth term, you'd plug in \(n = 10\), and get \(a_{10} = 4(10) - 7 = 33\).