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Chapter 12: Problem 8

Find the first four terms and the 100th term of the sequence. $$a_{n}=\frac{1}{n^{2}}$$

Short Answer

Expert verified
First four terms are 1, \( \frac{1}{4} \), \( \frac{1}{9} \), \( \frac{1}{16} \); 100th term is \( \frac{1}{10000} \).

Step by step solution

01

Understand the Sequence Formula

The sequence is defined by the formula \( a_n = \frac{1}{n^2} \). This means the nth term of the sequence is the reciprocal of the square of \( n \).
02

Calculate the First Term

Substitute \( n = 1 \) into the formula: \[a_1 = \frac{1}{1^2} = 1\]So, the first term \( a_1 \) is 1.
03

Calculate the Second Term

Substitute \( n = 2 \) into the formula: \[a_2 = \frac{1}{2^2} = \frac{1}{4}\]So, the second term \( a_2 \) is \( \frac{1}{4} \).
04

Calculate the Third Term

Substitute \( n = 3 \) into the formula: \[a_3 = \frac{1}{3^2} = \frac{1}{9}\]So, the third term \( a_3 \) is \( \frac{1}{9} \).
05

Calculate the Fourth Term

Substitute \( n = 4 \) into the formula: \[a_4 = \frac{1}{4^2} = \frac{1}{16}\]So, the fourth term \( a_4 \) is \( \frac{1}{16} \).
06

Calculate the 100th Term

Substitute \( n = 100 \) into the formula: \[a_{100} = \frac{1}{100^2} = \frac{1}{10000}\]So, the 100th term \( a_{100} \) is \( \frac{1}{10000} \).

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Key Concepts

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

Understanding the Sequence Formula
A sequence formula act as a blueprint in mathematics to determine each term of a sequence. In this exercise, the sequence formula is given by \(a_n = \frac{1}{n^2}\). Here, \(n\) is a positive integer, which denotes the position of a term in the sequence. This formula effectively tells us that each term, \(a_n\), is calculated by taking the reciprocal of the square of \(n\).
  • Sequential formulas help in predicting terms without listing them all.
  • They are crucial for identifying patterns in sequences.
In the given formula, \(\frac{1}{n^2}\) reveals a pattern where fractions become smaller as \(n\) increases, showcasing how sequences converge or diverge.
The Concept of Reciprocal
The term "reciprocal" refers to the inverse of a number. Basically, if you have a number \(x\), its reciprocal is \(\frac{1}{x}\). For any non-zero real number, multiplying it by its reciprocal always results in 1.
In our sequence formula, \(\frac{1}{n^2}\) uses the reciprocal of the square of \(n\), meaning we're flipping the fraction. Hence:
  • For \(n = 1\), the reciprocal of \(1^2\) is \(\frac{1}{1} = 1\).
  • For \(n = 2\), the reciprocal of \(4\) is \(\frac{1}{4}\).
  • For \(n = 3\), the reciprocal of \(9\) is \(\frac{1}{9}\).
Reciprocals are important in deriving terms from formulae, ensuring effective computation of values in sequences.
Understanding the Square of a Number
Squaring is an arithmetic operation where a number is multiplied by itself. For instance, the square of \(n\) is given by \(n^2 = n \times n\). Squaring results in a positive number because:
  • negative times negative yields positive, as seen when squaring negative numbers.
In the context of our sequence, the formula \(a_n = \frac{1}{n^2}\) involves squaring \(n\) before finding the reciprocal:
  • \(n = 1\), \(1^2 = 1\)
  • \(n = 2\), \(2^2 = 4\)
  • \(n = 3\), \(3^2 = 9\)
Squaring forms the base of derivation in many algebraic and arithmetic computations, significantly impacting our sequence formula by determining the denominator.
What Are Terms of a Sequence?
In mathematics, a sequence is an ordered set of numbers, and each of these numbers is called a term. Terms occupy specific positions in a sequence and are typically designated by \(a_n\), where \(n\) is an index representing the term's position.
In our exercise, the sequence is specified by \(a_n = \frac{1}{n^2}\):
  • The first term, \(a_1\), is \(1\).
  • The second term, \(a_2\), is \(\frac{1}{4}\).
  • The third term, \(a_3\), is \(\frac{1}{9}\).
  • The hundredth term, \(a_{100}\), is \(\frac{1}{10000}\).
Terms of a sequence are crucial for analyzing patterns and mathematical behaviors. Recognizing patterns can simplify understanding and predicting future terms within the sequence.

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