Problem 8 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \( n \).

Step 2: 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.

Step 3: 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 \( \frac{1}{4} \).

Step 4: 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 \( \frac{1}{9} \).

Step 5: 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 \( \frac{1}{16} \).

Step 6: 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 \( \frac{1}{10000} \).

Key Concepts

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.