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Magazine Chapter 13: Problem 15
Write the first five terms of each sequence. $$ a_{n}=5(-1)^{n-1} $$
Short Answer
Expert verified
5, -5, 5, -5, 5
Step by step solution
01
Understand the General Formula
The general formula for the sequence is given by: a_{n} = 5(-1)^{n-1}This formula will be used to compute each term in the sequence.
02
Compute the First Term (n = 1)
Substitute n = 1 into the general formula: a_{1} = 5(-1)^{1-1} = 5(-1)^{0} = 5(1) = 5
03
Compute the Second Term (n = 2)
Substitute n = 2 into the general formula: a_{2} = 5(-1)^{2-1} = 5(-1)^{1} = 5(-1) = -5
04
Compute the Third Term (n = 3)
Substitute n = 3 into the general formula: a_{3} = 5(-1)^{3-1} = 5(-1)^{2} = 5(1) = 5
05
Compute the Fourth Term (n = 4)
Substitute n = 4 into the general formula: a_{4} = 5(-1)^{4-1} = 5(-1)^{3} = 5(-1) = -5
06
Compute the Fifth Term (n = 5)
Substitute n = 5 into the general formula: a_{5} = 5(-1)^{5-1} = 5(-1)^{4} = 5(1) = 5
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the General Formula
The general formula is like a recipe for making each term in a sequence. In this exercise, the general formula is given by: \( a_{n} = 5(-1)^{n-1} \). This formula tells us how to compute any term in the sequence by simply plugging in the value of \( n \) (which is the position of the term in the sequence) into the formula.In our formula, \( 5 \) is a constant that stays the same for all terms. The part that changes is \( (-1)^{n-1} \). This part makes the sequence alternate between positive and negative values. The exponent \( n-1 \) means we are using the term's position (n) minus one. By understanding these two parts, we can easily compute any term in the sequence.
Step-by-Step Term Computation
Now, let's break down how to compute each term using the general formula. First, we'll substitute the term number (n) into the formula.
- Term 1 (n = 1): Substitute \( n = 1 \) into the formula: \( a_{1} = 5(-1)^{1-1} = 5(-1)^{0} = 5(1) = 5 \). So the first term is 5.
- Term 2 (n = 2): Substitute \( n = 2 \) into the formula: \( a_{2} = 5(-1)^{2-1} = 5(-1)^{1} = 5(-1) = -5 \). The second term is -5.
- Term 3 (n = 3): Substitute \( n = 3 \) into the formula: \( a_{3} = 5(-1)^{3-1} = 5(-1)^{2} = 5(1) = 5 \). The third term is 5.
- Term 4 (n = 4): Substitute \( n = 4 \) into the formula: \( a_{4} = 5(-1)^{4-1} = 5(-1)^{3} = 5(-1) = -5 \). The fourth term is -5.
- Term 5 (n = 5): Substitute \( n = 5 \) into the formula: \( a_{5} = 5(-1)^{5-1} = 5(-1)^{4} = 5(1) = 5 \). The fifth term is 5.
Recognizing an Alternating Sequence
An alternating sequence is a sequence where the signs of the terms keep flipping between positive and negative. In our given formula\( a_{n} = 5(-1)^{n-1} \), the alternation is controlled by \( (-1)^{n-1} \). Here's what happens:
- When \( n \) is odd, \( (-1) \) raised to an even power (because \( n-1 \)) is even, becomes positive 1.
- When \( n \) is even, \( (-1) \) raised to an odd power (because \( n-1 \)) is odd, becomes negative 1.
- In positions 1, 3, and 5, the terms are all positive (5, 5, 5).
- In positions 2 and 4, the terms are negative (-5, -5).
Introduction to Mathematical Sequences
A sequence is basically an ordered list of numbers following a specific pattern. In mathematics, sequences are usually described by a formula that allows you to find any term in the sequence. This exercise features a type of sequence known as an **alternating sequence**, where the signs of the terms switch between positive and negative. Let's sum up the key points here:
- General Formula: Describes how to calculate any term in a sequence. For our sequence, \( a_{n} = 5(-1)^{n-1} \) is the rule for finding terms.
- Term Computation: Involves substituting the term's position into the general formula.
- Alternating Sequence: A sequence where signs of terms alternate (positive, negative, positive, and so on).
