Problem 10 - Precalculus: Mathematics for Calculus

Question

Find the first four terms and the 100 th term of the sequence whose \(n\)th term is given. \(a_{n}=\frac{1}{n^{2}}\)

Step-by-Step Solution

Answer

First four terms: 1, 1/4, 1/9, 1/16; 100th term: 1/10000.

Step 1: Identify the Formula

The sequence is defined by the term \(a_n = \frac{1}{n^2}\). This formula will be used to calculate each specific term of the sequence.

Step 2: Calculate the First Term

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

Step 3: Calculate the Second Term

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

Step 4: Calculate the Third Term

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

Step 5: Calculate the Fourth Term

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

Step 6: Calculate the 100th Term

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

Key Concepts

Sequence Terms Calculation

When you come across a sequence, each term within it has a specific position. Calculating these terms correctly is fundamental to understanding sequences. In the sequence given by the formula \(a_n = \frac{1}{n^2}\), each term is determined by substituting the position number, \(n\), into this formula. Let's delve into this process of substitution for sequence terms calculation.

For the calculation of specific terms, you begin by identifying the value of \(n\) for that position:

• For the first term, \(n = 1\), calculate it by plugging \(n\) into the formula: \(a_1 = \frac{1}{1^2} = 1\).
• Similarly, for the second term with \(n = 2\): \(a_2 = \frac{1}{2^2} = \frac{1}{4}\).
• The third term where \(n = 3\): \(a_3 = \frac{1}{3^2} = \frac{1}{9}\).
• The fourth term with \(n = 4\): \(a_4 = \frac{1}{4^2} = \frac{1}{16}\).

These steps continue for as many terms as needed, each time adjusting \(n\) to find the next term. This approach helps in organizing and identifying each sequence term clearly.

nth Term Formula

The \(n\)th term formula of a sequence provides a standard way to describe the terms of a sequence. It allows you to find any term without writing out the entire sequence. This is particularly helpful for large sequences.

In our case, the \(n\)th term formula is expressed as \(a_n = \frac{1}{n^2}\). This formula tells us that the value of each term in the series is calculated by taking the reciprocal of the square of its position number \(n\).

Using this formula:

• The formula provides a direct computation method without needing previous numbers. If you need the 100th term, you simply substitute \(n = 100\) into the formula \(a_{100} = \frac{1}{100^2} = \frac{1}{10000}\).
• This formula applies universally within the sequence, regardless of the term number.

Thus, having an \(n\)th term formula simplifies the process of sequence evaluation significantly, whether for early or late terms.

Arithmetic Sequence

It's important to distinguish different types of sequences in mathematics. The given sequence \(a_n = \frac{1}{n^2}\) appears more similar to a sequence of inverse squares rather than an arithmetic sequence.

An arithmetic sequence is defined by a constant difference between consecutive terms. In contrast, our sequence involves powers and reciprocals, leading to non-uniform differences. For example:

• Moving from the first term \(1\) to the second term \(\frac{1}{4}\) gives a difference of \(\frac{3}{4}\).
• Transitioning from the second term \(\frac{1}{4}\) to the third term \(\frac{1}{9}\) results in a difference of \(\frac{5}{36}\).

Such variations indicate it is not arithmetic.

This example highlights the importance of understanding the nature of a sequence, allowing you to apply the correct calculations and formulas suitable for each kind of sequence.