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Magazine Chapter 13: Problem 46
\(43-48\) . Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$ a=100, d=-5, n=8 $$
Short Answer
Expert verified
The partial sum of the sequence is 660.
Step by step solution
01
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 \(n = 8\).
02
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: \( 100, 95, 90, 85, 80, 75, 70, \, \text{and} \, 65 \).
03
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 \( n = 8 \).
04
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 \[ S_8 = 4 (200 - 35) \].
05
Solving the Equation
Now compute the simplified equation: \[ S_8 = 4 \times 165 = 660 \]. Thus, the partial sum of the sequence is 660.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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鈥攁 "common difference"鈥攖o the previous term. To find a partial sum, you don't need to manually add each term; instead, there's a handy formula:
- \( S_n = \frac{n}{2} (2a + (n-1)d) \)
- \( n \) is the number of terms you want to sum up
- \( a \) is the first term
- \( d \) is the common difference
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鈥擻( 100, 95, 90, 85, 80, 75, 70, 65 \)鈥攖he common difference is \( -5 \).
- In the sequence from the exercise鈥擻( 100, 95, 90, 85, 80, 75, 70, 65 \)鈥攖he common difference is \( -5 \).
- This means each term is 5 units less than the term before it.
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 \( a = 100 \).
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.
