Problem 46 - Algebra and Trigonometry

Question

\(43-48\) . Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$ a=100, d=-5, n=8 $$

Step-by-Step Solution

Answer

The partial sum of the sequence is 660.

Step 1: Understanding the Problem

We are tasked with finding the partial sum, \(S_n\), of an arithmetic sequence given the first term \(a = 100\), the common difference \(d = -5\), and the number of terms \(n = 8\).

Step 2: Form of Arithmetic Sequence

An arithmetic sequence is defined as \( a, a+d, a+2d, \, \ldots, \, a+(n-1)d \). Hence, for our sequence, the terms are: \( 100, 95, 90, 85, 80, 75, 70, \, \text{and} \, 65 \).

Step 3: Using the Formula for Partial Sum of Arithmetic Sequence

The formula for the partial sum of an arithmetic sequence is \( S_n = \frac{n}{2} (2a + (n-1)d) \). For our sequence, we substitute \( a = 100 \), \( d = -5 \), and \( n = 8 \).

Step 4: Substituting Values into the Formula

Substitute the given values into the formula: \[ S_8 = \frac{8}{2} (2 \times 100 + (8-1) \times -5) \]. This simplifies to \[ S_8 = 4 (200 - 35) \].

Step 5: Solving the Equation

Now compute the simplified equation: \[ S_8 = 4 \times 165 = 660 \]. Thus, the partial sum of the sequence is 660.

Key Concepts

Partial Sum

The concept of a "partial sum" in an arithmetic sequence is quite simple once you get the hang of it. The partial sum, often denoted as \( S_n \), is the total sum of the first \( n \) terms of an arithmetic sequence. An arithmetic sequence is a series of numbers in which each term after the first is found by adding a fixed and consistent amount—a "common difference"—to the previous term. To find a partial sum, you don't need to manually add each term; instead, there's a handy formula:

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

Let's break it down:

• \( n \) is the number of terms you want to sum up
• \( a \) is the first term
• \( d \) is the common difference

This formula ensures you can quickly compute the total sum without error, just as we did in the original problem by finding that \( S_8 \) equals 660.

Common Difference

In every arithmetic sequence, there is an essential feature called the "common difference." The common difference, indicated by \( d \), is the amount that each term in the sequence increases or decreases from the previous term. It is called "common" because it remains the same throughout the sequence.For example:
- In the sequence from the exercise—\( 100, 95, 90, 85, 80, 75, 70, 65 \)—the common difference is \( -5 \).

• This means each term is 5 units less than the term before it.

The common difference is crucial for accurately continuing or finding the sums of the sequence. You simply take any term and subtract the previous term to find \( d \). Be sure to keep the sign in mind; positive means increasing, and negative means decreasing.

First Term

The first term of an arithmetic sequence is a critical starting point. It is typically represented by \( a \). This term forms the basis from which all subsequent terms are calculated using the common difference.In the example given:

• The first term is \( a = 100 \).

When you construct an arithmetic sequence, beginning with this first term and applying the common difference repeatedly will generate the entire sequence. It's like setting the tone for the rest of your series—choose wisely, as it impacts every subsequent term.The first term is our anchor in the formula for partial sums because it's always the starting point for calculations. Without the first term, the entire sequence would have no foundation.

Number of Terms

The "number of terms" in an arithmetic sequence is represented by \( n \). This value tells you how many terms from the sequence you are interested in. It's pivotal for calculating things like partial sums.In our problem:

• We want to find the sum of the first \( n = 8 \) terms.

Using \( n \), you determine how many terms look beyond just the starting point and common difference. This component allows you to decide how extensive your sequence analysis will be and is a key parameter in the partial sum formula, directly affecting the sum outcome as larger \( n \) means more terms to add up.Whenever you calculate or work with sequences, \( n \) must be clearly understood to achieve accurate results.