Problem 13 - College Algebra

Question

Write the first five terms of the arithmetic series given two terms. $$ a_{13}=-60, a_{33}=-160 $$

Step-by-Step Solution

Answer

The first five terms are: 0, -5, -10, -15, -20.

Step 1: Understand the Terms Given

We are given two terms of the arithmetic series, specifically the 13th term \(a_{13} = -60\) and the 33rd term \(a_{33} = -160\). In an arithmetic series, the general term can be defined as \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term and \(d\) is the common difference.

Step 2: Set Up Equations for Given Terms

Based on the formula \(a_n = a_1 + (n-1)d\), set up equations for \(a_{13}\) and \(a_{33}\): 1. \(-60 = a_1 + 12d\) 2. \(-160 = a_1 + 32d\).

Step 3: Solve for the Common Difference

Subtract the first equation from the second equation to eliminate \(a_1\): \(-160 - (-60) = (a_1 + 32d) - (a_1 + 12d)\).This simplifies to \(-100 = 20d\). Solve for \(d\): \(d = \frac{-100}{20} = -5\).

Step 4: Solve for the First Term

Substitute the value of \(d\) back into either equation from Step 2. Let's use the first equation: \(-60 = a_1 + 12(-5)\).Simplify to find \(a_1\): \(-60 = a_1 - 60\) which gives \(a_1 = 0\).

Step 5: Calculate the First Five Terms of the Series

Use the formula \(a_n = a_1 + (n-1)d\) where \(a_1 = 0\) and \(d = -5\): 1. \(a_1 = 0\)2. \(a_2 = 0 + 1(-5) = -5\)3. \(a_3 = 0 + 2(-5) = -10\)4. \(a_4 = 0 + 3(-5) = -15\)5. \(a_5 = 0 + 4(-5) = -20\).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is obtained by adding a constant value, called the "common difference," to the previous term. This sequence is also referred to as an arithmetic progression. It's essential in understanding the systematic approach to listing numbers in a specific pattern.

In other words, if you start with a number called the first term and keep adding the same number over and over, you form an arithmetic sequence. This property makes it much easier to predict future terms and analyze patterns.

• For example, if you think about the sequence 2, 5, 8, 11, ..., each term is 3 more than the one before it. Here, 3 is the common difference.

This concept is a stepping stone for learning about other sequences and series and is foundational in mathematical concepts like calculus and beyond.

Common Difference

The common difference in an arithmetic sequence is the fixed amount you add (or subtract if it’s negative) to get from one term to the next. It's like the secret ingredient that keeps the sequence in order and is denoted by the letter \(d\).

Understanding how to find the common difference is crucial because it determines the step size between terms in the sequence. For example, say you have a sequence where the first term is 7 and the common difference is 3. The next few terms would be 10, 13, 16, and so on, illustrating how the sequence grows.

• Finding the common difference involves simply subtracting one term from the subsequent term in the sequence.
• In the problem above, since the difference between the terms \(a_{33} = -160\) and \(a_{13} = -60\) was given over 20 intervals, the common difference could be calculated as \(d = \frac{-100}{20} = -5\).

Learning to calculate the common difference quickly can save you time and ensure accuracy when dealing with arithmetic sequences.

First Term of Sequence

The first term of an arithmetic sequence, often denoted as \(a_1\), is the starting point of the sequence. It's an important piece that, along with the common difference, helps define the entire sequence.

Let's think of it as the "anchor" of the sequence. Without knowing the first term, you wouldn't know where your sequence begins. This term is a key in formulating the general expression for any term in the sequence, \(a_n = a_1 + (n-1)d\), which describes how to find any term in a sequence given the first term and the common difference.

• In the exercise provided, solving for the first term using the equation \(-60 = a_1 + 12(-5)\) gives \(a_1 = 0\), showing how to pinpoint the starting number of your sequence.

Understanding how to identify or compute the first term ensures you start a sequence correctly, laying a foundation for accurately finding subsequent terms. This understanding helps solidify your grasp of how arithmetic sequences work and their algebraic formulation.