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Chapter 9: Problem 15

Find a formula for the \(n\)th term of the sequence. The sequence \(1,-4,9,-16,25, \ldots\)

Short Answer

Expert verified
The formula for the nth term is: \( a_n = (-1)^{n+1} n^2 \).

Step by step solution

01

Identify the Sequence Pattern

The given sequence is: 1, -4, 9, -16, 25, ... Observe the sequence of numbers: the absolute values of the sequence are 1, 4, 9, 16, 25, ... which are perfect squares. Specifically, these terms are 1^2, 2^2, 3^2, 4^2, and 5^2 respectively.
02

Determine Sign Pattern

The sequence alternates signs: positive, negative, positive, negative, positive, ... This can be represented by the factor (-1) raised to the power n+1, because (-1)^{n+1} gives 1 for odd n and -1 for even n.
03

Combine Patterns

Combining both the square number pattern and the sign pattern, a formula can be established. The nth term is given by the square of n with an alternating sign factor, leading to the general formula: a_n = (-1)^{n+1} imes n^2.

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Key Concepts

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

Number Patterns
The concept of number patterns is foundational in identifying sequences and series. A number pattern is a sequence of numbers that follows a particular rule or formula, allowing us to predict subsequent numbers in the sequence.

In this exercise, the given sequence is \(1, -4, 9, -16, 25, \ldots\). Each term in this sequence follows a particular number pattern, which makes it easier to understand and calculate.
  • The sequence's absolute values correspond to perfect squares: \(1, 4, 9, 16, 25, \ldots\).
  • This reveals that the terms \(1^2, 2^2, 3^2, 4^2, 5^2\) produce the observed numbers, \(1, 4, 9, 16, 25\).
By identifying this pattern, we can deduce that the nth term of the absolute values is \(n^2\). Recognizing these patterns is crucial for breaking down complex sequences into simpler parts for analysis.
Alternating Sequences
Alternating sequences involve terms that change between two distinct values or states. In mathematics, this often pertains to a change in sign, such as alternating between positive and negative values.

The sequence in the exercise shows an alternating sign pattern.
  • The signs of the sequence are: positive, negative, positive, negative, positive, \ldots\
  • This forms an alternation between \(+1\) and \(-1\) for successive terms.
To denote this mathematically, we use the expression \((-1)^{n+1}\). This expression takes advantage of the property that \((-1)\) raised to an even power is \(1\), and raised to an odd power is \(-1\). Consequently, this expression provides a concise way to alternate signs directly tied to the position \(n\) of the term in the sequence.
Perfect Squares
Perfect squares are numbers that are the square of an integer. They play an important role in identifying and constructing number sequences. A perfect square can be written as \(n^2\), where \(n\) is a whole number.

In the given sequence, the absolute values of the terms are perfect squares.
  • The first term is \(1\) or \(1^2\).
  • The second term is \(4\) or \(2^2\).
  • The third term is \(9\) or \(3^2\), and so forth.
This pattern of perfect squares helps in forming the rule or formula for the sequence. By recognizing that the absolute value of each term corresponds to \(n^2\), we can combine this with the sign alteration pattern to construct the entire sequence formula \(a_n = (-1)^{n+1} \times n^2\). This formula effectively encapsulates both the alternating sign and the pattern of perfect squares.

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Most popular questions from this chapter

Uniqueness of least upper bounds Show that if \(M_{1}\) and \(M_{2}\) are least upper bounds for the sequence \(\left\\{a_{n}\right\\},\) then \(M_{1}=M_{2}\) That is, a sequence cannot have two different least upper bounds.

Which of the sequences converge, and which diverge? Give reasons for your answers. $$a_{n}=1-\frac{1}{n}$$

The zipper theorem \(\quad\) Prove the "zipper theorem" for sequences: If \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) both converge to \(L\), then the sequence $$a_{1}, b_{1}, a_{2}, b_{2}, \ldots, a_{n}, b_{n}, \ldots$$ converges to \(L\).

For what values of \(r\) does the infinite series $$1+2 r+r^{2}+2 r^{3}+r^{4}+2 r^{5}+r^{6}+\cdots$$ converge? Find the sum of the serics when it converges.

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit \(L ?\) b. If the sequence converges, find an integer \(N\) such that \(\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$a_{n}=\frac{\ln n}{n}$$
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