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Magazine Chapter 13: Problem 16
Write the first five terms of each sequence. $$ a_{n}=6(-1)^{n+1} $$
Short Answer
Expert verified
The first five terms are 6, -6, 6, -6, 6.
Step by step solution
01
Understand the Sequence Formula
The sequence is given by the formula \( a_{n} = 6(-1)^{n+1} \). This means that each term depends on the value of \( n \) and alternates in sign due to \((-1)^{n+1}\).
02
Calculate the First Term \( a_{1} \)
Substitute \( n = 1 \) into the formula: \( a_{1} = 6(-1)^{1+1} = 6(-1)^{2} = 6(1) = 6 \). So, the first term is 6.
03
Calculate the Second Term \( a_{2} \)
Substitute \( n = 2 \) into the formula: \( a_{2} = 6(-1)^{2+1} = 6(-1)^{3} = 6(-1) = -6 \). So, the second term is -6.
04
Calculate the Third Term \( a_{3} \)
Substitute \( n = 3 \) into the formula: \( a_{3} = 6(-1)^{3+1} = 6(-1)^{4} = 6(1) = 6 \). So, the third term is 6.
05
Calculate the Fourth Term \( a_{4} \)
Substitute \( n = 4 \) into the formula: \( a_{4} = 6(-1)^{4+1} = 6(-1)^{5} = 6(-1) = -6 \). So, the fourth term is -6.
06
Calculate the Fifth Term \( a_{5} \)
Substitute \( n = 5 \) into the formula: \( a_{5} = 6(-1)^{5+1} = 6(-1)^{6} = 6(1) = 6 \). So, the fifth term is 6.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
sequence formula
In algebra, a sequence is a list of numbers arranged in a specific order following a specific rule. A sequence formula allows us to find any term in the sequence without having to list all previous terms. For instance, consider the formula given: \( a_{n} = 6(-1)^{n+1} \). This formula informs us about the value of the \( n \text{th} \) term of the sequence. Here, \( n \) represents the position of the term in the sequence. The sequence alternates in sign because of the \((-1)^{n+1}\) part.
term calculation
To calculate the terms of a sequence with the formula \( a_{n} = 6(-1)^{n+1} \), you simply substitute different values of \( n \) into the formula:
- First Term (\( a_{1} \)): Substitute \( n = 1 \) into the formula to get \( a_{1} = 6(-1)^{1+1} = 6(1) = 6 \)
- Second Term (\( a_{2} \)): Substitute \( n = 2 \) into the formula to get \( a_{2} = 6(-1)^{2+1} = 6(-1) = -6 \)
- Third Term (\( a_{3} \)): Substitute \( n = 3 \) into the formula to get \( a_{3} = 6(-1)^{3+1} = 6(1) = 6 \)
- Fourth Term (\( a_{4} \)): Substitute \( n = 4 \) into the formula to get \( a_{4} = 6(-1)^{4+1} = 6(-1) = -6 \)
- Fifth Term (\( a_{5} \)): Substitute \( n = 5 \) into the formula to get \( a_{5} = 6(-1)^{5+1} = 6(1) = 6 \)
alternating sign sequence
An alternating sign sequence is a sequence in which the signs of the terms alternate between positive and negative. In the example provided (\( a_{n} = 6(-1)^{n+1} \)), the term alternates every time \( n \) changes. This means that:
- Terms with odd \( n \) (like \( a_{1}, a_{3}, a_{5}, \text{etc.} \)) are positive.
- Terms with even \( n \) (like \( a_{2}, a_{4}, \text{etc.} \)) are negative.
mathematical patterns
Mathematical patterns in sequences help us predict the future terms without calculation. Patterns can be numerical, visual, or based on a rule like alternation. In our given sequence formula \( a_{n} = 6(-1)^{n+1} \), we see a clear numerical and sign pattern:
- Numeric constant: Each term is either \( 6 \) or \( -6 \).
- Alternating sign: The sign changes with each subsequent term.
