Problem 8 - Algebra and Trigonometry

Question

\(3-12\) . Find the first four terms and the 100 th term of the sequence. $$ a_{n}=\frac{1}{n^{2}} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Sequence Formula

The given formula for the sequence is \( a_n = \frac{1}{n^2} \). This means that each term in the sequence is the reciprocal of the square of its position number \( n \). Our job is to find specific terms of this sequence: the first four terms and the 100th term.

Step 2: Calculating the First Term

To find the first term of the sequence, substitute \( n = 1 \) into the formula. \( a_1 = \frac{1}{1^2} = 1 \). So, the first term is 1.

Step 3: Calculating the Second Term

To find the second term of the sequence, substitute \( n = 2 \) into the formula. \( a_2 = \frac{1}{2^2} = \frac{1}{4} \). So, the second term is \( \frac{1}{4} \).

Step 4: Calculating the Third Term

To find the third term of the sequence, substitute \( n = 3 \) into the formula. \( a_3 = \frac{1}{3^2} = \frac{1}{9} \). So, the third term is \( \frac{1}{9} \).

Step 5: Calculating the Fourth Term

To find the fourth term of the sequence, substitute \( n = 4 \) into the formula. \( a_4 = \frac{1}{4^2} = \frac{1}{16} \). So, the fourth term is \( \frac{1}{16} \).

Step 6: Calculating the 100th Term

To find the 100th term of the sequence, substitute \( n = 100 \) into the formula. \( a_{100} = \frac{1}{100^2} = \frac{1}{10000} \). So, the 100th term is \( \frac{1}{10000} \).

Key Concepts

Arithmetic sequence

Arithmetic sequences are a type of sequence in mathematics where each subsequent term is derived by adding a constant value to the previous term. This constant value is known as the "common difference." Here's a simple way to think about it:

• If the first term of an arithmetic sequence is denoted as \( a_1 \), and the common difference is \( d \), the next term, \( a_2 \), will be \( a_1 + d \).
• Then, \( a_3 \) will be \( a_2 + d \), and this pattern continues.

We often see real-world examples of arithmetic sequences, such as the number of seats in each row of a theater if they increase by a fixed number with every row. However, not all sequences are arithmetic. In fact, the sequence given in the original exercise is not an arithmetic sequence, because its terms are derived using a reciprocal function, not by adding a constant to the previous term. Understanding this distinction is crucial for solving sequence-related problems correctly.

Formula for terms

In any sequence, a formula that defines the nth term is crucial for understanding the structure and pattern of the sequence. The formula provides a mathematical way to find any term you want without having to list all terms up to that point.

Consider the sequence in the original exercise: the formula is given by \( a_n = \frac{1}{n^2} \). Here, the nth term, \( a_n \), is found by taking the reciprocal of the square of \( n \). It is not an arithmetic or geometric formula, but a specific form related to reciprocal functions.

• The formula provides a quick method to calculate any term directly without computing all previous terms.
• This is particularly useful for large term numbers, such as the 100th term, which can be directly found without preceding calculations.

Understanding this concept allows you to work efficiently with sequences, finding specific or general terms quickly.

Reciprocal function

A reciprocal function is an important concept in mathematics, often forming the basis of various mathematical sequences. The reciprocal of a number \( x \) is simply \( \frac{1}{x} \), which flips the number over as a fraction.

In terms of sequences, using a reciprocal function can add unique characteristics to the sequence. The given sequence formula in the problem, \( a_n = \frac{1}{n^2} \), uses a reciprocal function where each term of the sequence is the reciprocal of the square of its position number:

• The first term is \( \frac{1}{1^2} = 1 \)
• The second term is \( \frac{1}{2^2} = \frac{1}{4} \)
• This pattern continues, changing as per the inverse square of the position in the sequence.

Reciprocal functions introduce different rates of change in sequences compared to more regular arithmetic sequences, providing interesting alternative mathematical models.

Term calculation

Calculating terms in a sequence is an essential skill in mathematics, allowing you to identify specific elements without listing all terms. For our purpose, the given sequence formula \( a_n = \frac{1}{n^2} \) helps us find specific terms efficiently.

Let's break this down step by step:

• To find the 1st term, substitute \( n = 1 \) into the formula: \( a_1 = \frac{1}{1^2} = 1 \).
• For the 2nd term, use \( n = 2 \): \( a_2 = \frac{1}{2^2} = \frac{1}{4} \).
• To calculate the 3rd term, use \( n = 3 \): \( a_3 = \frac{1}{3^2} = \frac{1}{9} \).
• Continue this pattern, for example, by calculating the 4th term as \( a_4 = \frac{1}{4^2} = \frac{1}{16} \).
• For the 100th term, \( n = 100 \), giving \( a_{100} = \frac{1}{100^2} = \frac{1}{10000} \).

These calculations show the practical use of a sequence's formula, allowing you to derive any term efficiently without recalculations from the beginning of the sequence.