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9. Sequences, Series, & Induction
Arithmetic Sequences
1:56 minutesProblem 7
Textbook Question
Write the first six terms of each arithmetic sequence. a1= 5/2, d = -1/2
Verified step by step guidance1
Identify the first term \(a_1\) and the common difference \(d\) of the arithmetic sequence. Here, \(a_1 = \frac{5}{2}\) and \(d = -\frac{1}{2}\).
Recall the formula for the \(n\)-th term of an arithmetic sequence: \(a_n = a_1 + (n-1)d\).
Calculate the second term \(a_2\) by substituting \(n=2\) into the formula: \(a_2 = \frac{5}{2} + (2-1) \times \left(-\frac{1}{2}\right)\).
Find the third term \(a_3\) by substituting \(n=3\): \(a_3 = \frac{5}{2} + (3-1) \times \left(-\frac{1}{2}\right)\).
Continue this process to find the fourth, fifth, and sixth terms by substituting \(n=4, 5, 6\) respectively into the formula \(a_n = \frac{5}{2} + (n-1) \times \left(-\frac{1}{2}\right)\).
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arithmetic Sequence
An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant difference to the previous term. This constant is called the common difference, and it defines the pattern of the sequence.
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Common Difference (d)
The common difference is the fixed amount added to each term to get the next term in an arithmetic sequence. It can be positive, negative, or zero, and it determines whether the sequence increases, decreases, or remains constant.
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Finding Terms of an Arithmetic Sequence
To find terms in an arithmetic sequence, use the formula a_n = a_1 + (n - 1)d, where a_1 is the first term, d is the common difference, and n is the term number. This formula helps calculate any term without listing all previous terms.
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