In mathematics, progression refers to a sequence of numbers arranged in a specific order according to a particular rule. One key type of progression is an arithmetic sequence, where each term after the first is found by adding a constant difference to the previous term. This consistent pattern makes it easy to identify and calculate terms within the sequence.
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The first term of an arithmetic sequence can be any number, and it sets the starting point for the entire progression.
The common difference in an arithmetic sequence can be positive, negative, or zero, affecting whether the sequence increases, decreases, or remains constant.
An arithmetic sequence can be finite or infinite; it will have a set number of terms in finite cases and continue indefinitely in infinite cases.
The sum of the first n terms of an arithmetic sequence can be calculated using the formula $$S_n = \frac{n}{2}(a_1 + a_n)$$ or $$S_n = \frac{n}{2}(2a_1 + (n - 1)d)$$.
Graphing an arithmetic sequence produces a straight line, indicating a linear relationship between the term number and its value.
Review Questions
How does the concept of common difference define an arithmetic sequence and how can it be used to identify subsequent terms?
The common difference is crucial as it establishes the pattern of an arithmetic sequence by determining how much each term increases or decreases from the previous term. To find subsequent terms, you simply add the common difference to the last known term. For example, if the first term is 3 and the common difference is 5, the second term would be 8, and the third would be 13. This consistent addition creates a predictable pattern in the progression.
Illustrate how to derive the n-th term formula for an arithmetic sequence and provide an example using specific values.
To derive the n-th term formula for an arithmetic sequence, start with the first term denoted as $$a_1$$ and add the common difference $$d$$ multiplied by one less than n. This gives us $$a_n = a_1 + (n - 1)d$$. For instance, if $$a_1 = 2$$ and $$d = 3$$, then for n = 5, we calculate $$a_5 = 2 + (5 - 1)3 = 2 + 12 = 14$$. This shows that understanding this formula allows for quick calculations of any term in the sequence.
Evaluate how understanding progressions can assist in solving real-world problems involving sequences and patterns.
Understanding progressions, particularly arithmetic sequences, can significantly aid in various real-world applications like budgeting or planning where consistent increments are involved. For instance, if someone saves a fixed amount of money each week, recognizing this saving pattern allows them to predict their total savings over time using the principles of arithmetic sequences. By applying these concepts, one can make informed financial decisions and better understand patterns in everyday life.
The fixed amount that is added to each term in an arithmetic sequence to get the next term.
n-th Term Formula: A formula used to find the n-th term of an arithmetic sequence, often expressed as $$a_n = a_1 + (n - 1)d$$, where $$a_1$$ is the first term and $$d$$ is the common difference.