Textbook Question
27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
c. Find an explicit formula for the nth term of the sequence.
{1, 2, 4, 8, 16, ......}
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14. Sequences & Series
Sequences
1:41 minutesProblem 10.1.33c
Textbook Question
27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
c. Find an explicit formula for the nth term of the sequence.
{-5, 5, -5, 5, ......}
Verified step by step guidance1
Identify the pattern in the given sequence: {-5, 5, -5, 5, ...}. Notice that the terms alternate between -5 and 5.
Recognize that this is an alternating sequence where the sign changes every term, but the absolute value remains constant at 5.
Recall that an alternating sign can be represented using powers of -1, specifically \((-1)^n\) or \((-1)^{n+1}\), which alternate between -1 and 1 as n increases.
Choose the appropriate power of -1 to match the first term: since the first term is -5 when n=1, test \((-1)^n\) and \((-1)^{n+1}\) to see which one gives -1 at n=1.
Write the explicit formula for the nth term as \(a_n = 5 \times (-1)^n\) or \(a_n = 5 \times (-1)^{n+1}\), depending on which matches the initial term, ensuring the sign alternates correctly.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sequences and Terms
A sequence is an ordered list of numbers defined by a specific pattern or rule. Each number in the sequence is called a term, denoted as aₙ, where n indicates the term's position. Understanding how terms change helps in identifying the sequence's behavior and formulating its general term.
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Introduction to Sequences
Explicit Formula for a Sequence
An explicit formula expresses the nth term of a sequence directly in terms of n, without needing previous terms. It allows calculation of any term independently. Finding this formula involves recognizing patterns such as arithmetic, geometric, or alternating signs in the sequence.
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Alternating Sequences and Sign Patterns
Alternating sequences switch signs between terms, often modeled using powers of -1. For example, (-1)ⁿ or (-1)ⁿ⁺¹ generates alternating positive and negative values. Identifying this pattern is key to writing an explicit formula for sequences like {-5, 5, -5, 5, ...}.
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