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Chapter 13: Problem 4

Fill in each blank with the correct response. If \(a_{n}=(-1)^{n}\), then \(a_{115}=\) ______.

Short Answer

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Step by step solution

01

Identify the expression for the sequence

The given sequence is defined by the expression \[ a_{n} = (-1)^{n} \]
02

Substitute the value of n

To find the value of the term at position 115, substitute \( n = 115 \) into the expression. Thus, \( a_{115} = (-1)^{115} \).
03

Simplify the expression

Recall that raising -1 to an odd power results in -1. Since 115 is an odd number, \[ (-1)^{115} = -1 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sequence Definition
In mathematics, a sequence is a list of numbers arranged in a particular order. Each number in the list is called a term. Sequences can be finite, having a fixed number of terms, or infinite, continuing indefinitely. One common type of sequence is the arithmetic sequence, where the difference between consecutive terms is constant. Another is the geometric sequence, where the ratio between consecutive terms is constant. For instance, the sequence \(a_{n}=(-1)^{n}\) is an infinite sequence that alternates between values -1 and 1 depending on whether the exponent \(n\) is even or odd.
Odd and Even Exponents
Exponents are used to indicate repeated multiplication of a number by itself. When the base is -1, the exponent plays a crucial role in determining the sign of the result.

Odd Exponents:
If the exponent is an odd number (e.g., 1, 3, 5, 7, ...), raising -1 to this power results in -1. This is because multiplying an odd number of negative ones gives a negative result. For example, \((-1)^{1}=-1\), \((-1)^{3}=-1\), and so on.

Even Exponents:
If the exponent is an even number (e.g., 2, 4, 6, 8, ...), raising -1 to this power results in 1. This is because multiplying an even number of negative ones gives a positive result. For example, \((-1)^{2}=1\), \((-1)^{4}=1\), and so on.
Sequence Term Calculation
To find a specific term in a sequence, you can substitute the term's position number into the sequence's explicit formula. For example, given the sequence \(a_{n}=(-1)^{n}\), you can find any term by substituting \(n\) with the position number.

Let's find the 115th term of the sequence:
\(a_{115}=(-1)^{115}\).
Since 115 is an odd number, according to the rules of exponents, \(a_{115}\) will be equal to -1. Therefore, \((-1)^{115} = -1\), so \(a_{115}=-1\).

Remember, understanding how exponents work will help you calculate terms easily.

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