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Chapter 11: Problem 3

Write the first five terms of each sequence, whose general term is given. See Example 1 $$ a_{n}=(-1)^{n} $$

Short Answer

Expert verified
The first five terms are: \(-1, 1, -1, 1, -1\).

Step by step solution

01

Understand the General Term

The general term given for the sequence is \(a_n = (-1)^n\). This means that the value of each term in the sequence depends on the parity of \(n\). When \(n\) is even, \((-1)^n = 1\), and when \(n\) is odd, \((-1)^n = -1\).
02

Calculate the First Term

To find the first term \(a_1\), substitute \(n = 1\) into the general term. Thus, \(a_1 = (-1)^1 = -1\).
03

Calculate the Second Term

To find the second term \(a_2\), substitute \(n = 2\) into the general term. Thus, \(a_2 = (-1)^2 = 1\).
04

Calculate the Third Term

To find the third term \(a_3\), substitute \(n = 3\) into the general term. Thus, \(a_3 = (-1)^3 = -1\).
05

Calculate the Fourth Term

To find the fourth term \(a_4\), substitute \(n = 4\) into the general term. Thus, \(a_4 = (-1)^4 = 1\).
06

Calculate the Fifth Term

To find the fifth term \(a_5\), substitute \(n = 5\) into the general term. Thus, \(a_5 = (-1)^5 = -1\).

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Key Concepts

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

General Term
In sequences in algebra, the general term is a formula that helps us find any term in the sequence just by substituting the position of the term.
In our case, we have the general term: \[ a_n = (-1)^n \].

This general term tells us that for each position \( n \), the value of the term is either \(1\) or \(-1\).
  • If \( n \) is an even number, \((-1)^n\) equals \(1\).
  • If \( n \) is an odd number, \((-1)^n\) equals \(-1\).
With this understanding, we can follow a clear pattern to determine each term. The general term is incredibly powerful as it allows us to calculate numerous terms further in the sequence without needing any prior terms.
Sequence Terms
The terms in a sequence are defined by substituting specific positions (indicated by \( n \)) into the general term.
The resulting values form the series of numbers called a sequence. For our sequence defined by \( a_n = (-1)^n \), here's how we calculate the first few terms:
  • For \( n = 1 \), the first term is \( a_1 = (-1)^1 = -1 \).
  • For \( n = 2 \), the second term is \( a_2 = (-1)^2 = 1 \).
  • For \( n = 3 \), the third term is \( a_3 = (-1)^3 = -1 \).
  • For \( n = 4 \), the fourth term is \( a_4 = (-1)^4 = 1 \).
  • For \( n = 5 \), the fifth term is \( a_5 = (-1)^5 = -1 \).
These sequence terms show alternating patterns of \(-1\) and \(1\), based on whether their position number \( n \) is odd or even.
Parity
Parity refers to whether an integer is odd or even.
This concept is directly related to how we interpret the general term \[ (-1)^n \].

  • When \( n \) is even, any whole number raised to an even power gives a positive result. Therefore, \((-1)^n\) will always equal \(1\).
  • Conversely, when \( n \) is odd, raising a whole number to an odd power maintains the negativity of the number. Thus, \((-1)^n\) will always equal \(-1\).
Understanding parity helps us predict sequence terms based on their position and recognize patterns within the sequences. It simplifies the process of determining which type of number to expect without manually calculating the term each time. Thus, parity plays a crucial role in sequences, especially in those defined by simple alternating patterns.

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Most popular questions from this chapter

In \(2007,\) the population of the Northern Spotted Owl continued to decline, and the owl remained on the endangered species list as old-growth Northwest forests were logged. The size of the decrease in the population in a given year can be estimated by \(200-6 n\) pairs of birds. Find the decrease in population in 2010 if year 1 is \(2007 .\) Find the estimated total decrease in the spotted owl population for the years 2007 through 2010 .

A person has a choice between two job offers. Job \(A\) has an annual starting salary of \(\$ 30,000\) with guaranteed annual raises of \(\$ 1200\) for the next four years, whereas job \(B\) has an annual starting salary of \(\$ 28,000\) with guaranteed annual raises of \(\$ 2500\) for the next four years. Compare the fifth partial sums for each sequence to determine which job would pay more money over the next 5 years.

A pendulum swings a length of 40 inches on its first swing. Each successive swing is \(\frac{4}{5}\) of the preceding swing. Find the length of the fifth swing and the total length swung during the first five swings. (Round to the nearest tenth of an inch.)

Solve. Find the sum of the first ten terms of the sequence \(-4,1,6, \ldots, 41\) where 41 is the tenth term.

Solve. See Example 2. Find the sum of the first four positive odd integers.
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