Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-22: Find the sum \(S_{n}\) of the arithmetic sequence that satisfies the stated conditions. $$ a_{1}=5, \quad d=0.1, \quad n=40 $$

Step-by-Step Solution

Answer

The sum of the arithmetic sequence is 278.

Step 1: Identify the formula for the sum of an arithmetic sequence

The sum of an arithmetic sequence can be calculated using the formula: \( S_{n} = \frac{n}{2} (a_{1} + a_{n}) \), where \( n \) is the number of terms, \( a_{1} \) is the first term, and \( a_{n} \) is the last term.

Step 2: Calculate the last term of the sequence

The last term of an arithmetic sequence \( a_{n} \) can be found using the formula: \( a_{n} = a_{1} + (n-1) \cdot d \). Substitute the given values: \( a_{1} = 5 \), \( d = 0.1 \), and \( n = 40 \). \[ a_{n} = 5 + (40-1) \cdot 0.1 = 5 + 3.9 = 8.9 \] So, \( a_{n} = 8.9 \).

Step 3: Substitute the values into the sum formula

We can now find the sum \( S_{n} \) using the formula: \[ S_{n} = \frac{40}{2} \times (5 + 8.9) = 20 \times 13.9 \].

Step 4: Calculate the sum

Perform the final calculation: \[ S_{n} = 20 \times 13.9 = 278 \] Therefore, the sum of the sequence is 278.

Key Concepts

Arithmetic Sequence Formula

An arithmetic sequence is a list of numbers with a very specific pattern: each number in the list is created by adding a constant value, called the common difference, to the previous number. To navigate this type of sequence, it's important to understand its building blocks.

• The first term is denoted as \(a_{1}\).
• The common difference, \(d\), is the amount added to each term to get to the next term.

To find the sum of the first \(n\) terms of an arithmetic sequence, we use the formula:\[S_{n} = \frac{n}{2} (a_{1} + a_{n})\]Here, \(a_{n}\) represents the last term of the sequence. This formula helps in calculating large sums quickly by leveraging the symmetry of the sequence. Understanding how to plug values into this formula is crucial for solving problems involving arithmetic sequences.

Sequence Sum Calculation

Calculating the sum of a sequence involves understanding both the structure of the sequence and the specifics of each term. Let's break down the steps once you have the formula:
First, make sure you have all necessary information:

• The number of terms \(n\): how many elements you are summing up.
• The first term \(a_{1}\): it sets the starting point of your series.
• The last term \(a_{n}\): which you need to calculate if not already given.

Now, substitute these values into the sum formula \(S_{n} = \frac{n}{2} (a_{1} + a_{n})\). As an example:

• If \(n = 40\), \(a_{1} = 5\), and \(a_{40} = 8.9\), then:
• Calculate the sum as \(S_{40} = \frac{40}{2} \times (5 + 8.9) = 278\).

This is the total sum of all 40 terms in the sequence. It's a very efficient way to calculate sums without adding each number individually.

Finding Last Term of Sequence

The last term of an arithmetic sequence can be found with a simple yet important formula:\[a_{n} = a_{1} + (n-1) \cdot d\]This formula helps you find \(a_{n}\), the nth term when you know:

• \(a_{1}\): the first term,
• \(d\): the common difference, and
• \(n\): the total number of terms.

For instance, with our problem:

• When \(a_{1} = 5\), \(d = 0.1\), and \(n = 40\), the last term is calculated as:
• \(a_{40} = 5 + (40-1) \times 0.1 = 8.9\).

Finding the last term is integral because it allows us to use the sequence sum formula, making it possible to calculate large sums efficiently.