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Magazine Chapter 9: Problem 13
For the following exercises, write the first five terms of the arithmetic series given two terms. $$ a_{13}=-60, a_{33}=-160 $$
Short Answer
Expert verified
The first five terms are 0, -5, -10, -15, -20.
Step by step solution
01
Identify Terms and Indices
We are given two terms of an arithmetic series: \(a_{13} = -60\) and \(a_{33} = -160\). Here, \(a_{13}\) is the 13th term and \(a_{33}\) is the 33rd term. We need to write the first five terms of this series.
02
Use the Formula for an Arithmetic Series
The formula for the \(n\)-th term of an arithmetic series is \(a_n = a_1 + (n-1) \cdot d\), where \(a_1\) is the first term and \(d\) is the common difference. We use this formula to express \(a_{13}\) and \(a_{33}\) in terms of \(a_1\) and \(d\).
03
Set up Equations from Given Terms
Using the formula, we have two equations: 1. \(a_1 + 12d = -60\) for the 13th term. 2. \(a_1 + 32d = -160\) for the 33rd term. We will use these equations to find \(a_1\) and \(d\).
04
Solve the System of Equations
Subtract the first equation from the second to eliminate \(a_1\): \((a_1 + 32d) - (a_1 + 12d) = -160 + 60\) Resulting in: \(20d = -100\) Thus, \(d = -5\).
05
Determine the First Term
Substitute \(d = -5\) back into the first equation: \(a_1 + 12(-5) = -60\) \(a_1 - 60 = -60\) Thus, \(a_1 = 0\).
06
Write the First Five Terms
Using the first term \(a_1 = 0\) and the common difference \(d = -5\), we calculate the first five terms: \(a_1 = 0\), \(a_2 = 0 - 5 = -5\), \(a_3 = -5 - 5 = -10\), \(a_4 = -10 - 5 = -15\), \(a_5 = -15 - 5 = -20\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
nth term formula
The nth term formula is a powerful tool in understanding arithmetic series. For any arithmetic sequence, the formula to calculate the nth term, denoted as \(a_n\), is given by:
\[ a_n = a_1 + (n-1) \cdot d \]
Here, \(a_1\) is the first term of the sequence, and \(d\) is the common difference between consecutive terms. The variable \(n\) represents the position of the term in the sequence, essentially pointing to which term we are calculating.
This formula helps us understand the linear nature of arithmetic sequences by explicitly showing how each term is derived from the first term and the common difference.
\[ a_n = a_1 + (n-1) \cdot d \]
Here, \(a_1\) is the first term of the sequence, and \(d\) is the common difference between consecutive terms. The variable \(n\) represents the position of the term in the sequence, essentially pointing to which term we are calculating.
This formula helps us understand the linear nature of arithmetic sequences by explicitly showing how each term is derived from the first term and the common difference.
system of equations
In this exercise, we used a system of equations to find the values of the first term \(a_1\) and the common difference \(d\). This approach is necessary when we are given specific terms in an arithmetic series but need more information to discover other terms.
When you have two equations like
When you have two equations like
- \(a_1 + 12d = -60\)
- \(a_1 + 32d = -160\)
common difference
The common difference \(d\) in an arithmetic sequence is the amount by which each term increases or decreases from the previous term. It is a critical parameter that defines the sequence's progression.
For instance, in our exercise, once we solved the system of equations, we found that
\(d = -5\).
This signifies that each term is 5 less than the preceding one. Knowing the common difference allows us to efficiently use the nth term formula to predict any term in the sequence, illustrating why it's such a foundational concept.
For instance, in our exercise, once we solved the system of equations, we found that
\(d = -5\).
This signifies that each term is 5 less than the preceding one. Knowing the common difference allows us to efficiently use the nth term formula to predict any term in the sequence, illustrating why it's such a foundational concept.
first term
The first term \(a_1\) of an arithmetic series serves as the starting point of the sequence. It is from this term that all other terms are calculated using both the first term itself and the common difference.
In the problem at hand, we found that the first term was
\(a_1 = 0\).
This information, combined with the common difference \(d\), allowed us to determine the successive terms of the sequence. It's essential to grasp this concept as it plays a significant role in evaluating and understanding arithmetic sequences fully. Every sequence has its unique starting point that determines its direction and magnitude.
In the problem at hand, we found that the first term was
\(a_1 = 0\).
This information, combined with the common difference \(d\), allowed us to determine the successive terms of the sequence. It's essential to grasp this concept as it plays a significant role in evaluating and understanding arithmetic sequences fully. Every sequence has its unique starting point that determines its direction and magnitude.
