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Chapter 13: Problem 39

Write a formula for the general term of each infinite sequence. \(0,1,4,9,16, \dots\)

Short Answer

Expert verified
The general term formula is a_n = (n-1)^2.

Step by step solution

01

- Analyze the Sequence

Observe the given sequence: 0, 1, 4, 9, 16,... Notice that each term in the sequence can be related to the square of its position in the sequence.
02

- Identify the Pattern

Identify the relationship between the term's position, denoted as n (starting from n=1 for the first term), and the value of the term. Here, each term seems to be the square of n-1.
03

- Write the General Term

Write the general term formula based on your observations. If n is the position in the sequence starting from 1, the corresponding term is Therefore, the nth term is: a_n = (n-1)^2

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Key Concepts

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

infinite sequence
An infinite sequence is a sequence that continues endlessly. In mathematics, such sequences do not stop at a certain number but extend indefinitely.
Infinite sequences are often used to describe patterns or sets of numbers that go on forever.
For example, the sequence 0, 1, 4, 9, 16,... is an infinite sequence of square numbers. Each number in the sequence has a specific position, or index, and the sequence can be described using a general formula.
problem-solving steps
Approaching a problem systematically is key to finding the correct solution. Here are some steps to solve the problem of finding the general term of an infinite sequence:
  • Analyze the Sequence: Look at the given terms to find a pattern or relationship. For example, in the sequence 0, 1, 4, 9, 16,..., notice how each term grows as the sequence progresses.
  • Identify the Pattern: Determine how each term in the sequence relates to its position. In our example, each term is the square of its position minus one.
  • Write the General Term: Based on the pattern, write a formula that represents any term in the sequence. For the given sequence, the formula for the nth term is \(a_n = (n-1)^2\).
Using these steps can help break down complex problems into more manageable parts.
square numbers
Square numbers are the result of multiplying a number by itself. The sequence 0, 1, 4, 9, 16,... is a classic example of square numbers.
To better understand square numbers:
  • 0 is the square of 0: \(0^2 = 0\)
  • 1 is the square of 1: \(1^2 = 1\)
  • 4 is the square of 2: \(2^2 = 4\)
  • 9 is the square of 3: \(3^2 = 9\)
  • 16 is the square of 4: \(4^2 = 16\)
Square numbers grow larger much faster than the natural numbers, and each term in the sequence represents a square of its predecessor's position plus one. Recognizing this pattern can help in forming the general term for the sequence, enhancing problem-solving abilities in mathematics.

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Most popular questions from this chapter

Write the repeating decimal number \(0.24242424 \ldots\) as an infinite geometric series. Find the sum of the geometric series.

Write the first four terms of the infinite sequence whose nth term is given. \(a_{n}=\frac{(-1)^{2 n}}{n^{2}}\)

Find the sum of each series. $$\sum_{i=1}^{20} 3$$

Consider the sequence whose \(n\) th term is \(a_{n}=(0.999)^{n}\). a) Calculate \(a_{100}, a_{1000},\) and \(a_{10,000}\). b) What happens to \(a_{n}\) as \(n\) gets larger and larger?

The first two terms of the Fibonacci sequence are 0 and \(1 .\) Every term thereafter is the sum of the two previous terms. So the third term is \(1,\) the fourth term is 2 the fifth term is \(3,\) and the sixth term is \(5 .\) So the first 6 terms of the Fibonacci sequence are \(0,1,1,2,3,5\). a) Write the first 10 terms of the Fibonacci sequence. b) Find an application of the Fibonacci sequence by doing a search on the Internet.
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