Problem 62 - Algebra 2

Question

Find \(S_{n}\) for each arithmetic series described. . \(a_{1}=-5, d=3, n=14\)

Step-by-Step Solution

Answer

The sum of the series is 203.

Step 1: Understand the Given Parameters

The task is to find the sum of an arithmetic series, denoted as \(S_{n}\), with the following parameters: the first term \(a_1 = -5\), the common difference \(d = 3\), and the number of terms \(n = 14\).

Step 2: Identify the Formula for Sum of Arithmetic Series

The formula to find the sum of the first \(n\) terms of an arithmetic series is given by:\[S_n = \frac{n}{2} (2a_1 + (n-1)d)\]This formula uses the number of terms \(n\), the first term \(a_1\), and the common difference \(d\).

Step 3: Substitute Known Values into the Formula

Substitute \(a_1 = -5\), \(d = 3\), and \(n = 14\) into the sum formula:\[S_{14} = \frac{14}{2} (2(-5) + (14-1) \cdot 3)\]

Step 4: Simplify Inside the Parentheses

First, simplify the expression inside the parentheses:- Calculate \(2 \cdot (-5) = -10\)- Calculate \((14-1) = 13\) and then \(13 \cdot 3 = 39\)- Combine terms \((-10 + 39)\) to get \(29\).Thus, the expression inside the parentheses becomes \(29\).

Step 5: Calculate the Sum

Finish the calculation:\[S_{14} = \frac{14}{2} \cdot 29 = 7 \cdot 29 = 203\]Therefore, the sum of the first \(14\) terms is \(203\).

Key Concepts

sum of arithmetic series

To find the sum of an arithmetic series, we need a formula that neatly packages everything together. This isn't as daunting as it sounds. The sum of an arithmetic series is simply the total of all its terms. Using the formula: \[ S_n = \frac{n}{2} (2a_1 + (n-1)d) \]we can calculate this sum. The formula cleverly simplifies the process by using the concepts of the first term, the common difference, and the total number of terms.

• It first takes half of the total number of terms, \( \frac{n}{2} \).
• It then multiplies this by the sum of twice the first term and the product of the common difference with the number of terms minus one.

This might sound complicated, but think of it as a set pattern: taking averages and then scaling up. That's a big win because it quickly condenses a potentially long addition (when you could just be adding terms one by one). Let's break it down further in the following sections.

common difference

Understanding the common difference is crucial in identifying the unique pattern in an arithmetic series. An arithmetic series is essentially a sequence where the difference between consecutive terms is constant. This difference, represented by \(d\), is known as the common difference. Here's how it works:

• If the common difference \(d\) is positive, each term in the series increases as you move from one to the next. Essentially, the series is on an upward trend.
• If \(d\) is negative, the series decreases, meaning each term is smaller than the one before it, indicating a downward trend.

The common difference plays a pivotal role in shaping the entire series since it directs how every term relates to the previous one. For the given problem, with \(d = 3\), every term increases by 3 from the one before, making it a series that grows as it progresses.

first term

The first term of an arithmetic series forms the foundation upon which the entire series is built. In an arithmetic series, the first term is represented by \(a_1\). It's the starting point from which subsequent terms are derived. Consider the first term as the anchor that determines the position of all other terms in the series. With the series given:

• We have \(a_1 = -5\), indicating that our starting point takes on the value of \(-5\).
• This sets the tone for everything that follows, as each additional term relies on the first term and the common difference \(d\) for the progression.

Therefore, the first term is vital, not just as a beginning point, but as a part of the calculation for the sum, through its role in the arithmetic sum formula. It helps determine the initial conditions of the series and shapes the flow of numbers to come.

number of terms

The number of terms is essentially the count of the entries you want in your series. It is represented by \(n\). This parameter specifies up to where you need to sum the series. In the given exercise:

• We have \(n = 14\), meaning we need to calculate the sum of the first 14 terms.
• This directs how many terms will be considered when finding the series sum.

It's similar to setting a cap on how far you're willing to go in adding up the values. Having a clear count is crucial because the sum will differ drastically if you, for instance, only sum 10 or decide to sum 20 terms instead. Understanding \(n\) helps to snapshot how large a portion of the series we are considering in our calculation.