Problem 27 - Algebra and Trigonometry

Question

Find the \(n\)th term of a sequence whose first several terms are given. \(1, \frac{3}{4}, \frac{5}{9}, \frac{7}{16}, \frac{9}{25}, \dots\)

Step-by-Step Solution

Answer

The nth term is \( a_n = \frac{2n - 1}{n^2} \).

Step 1: Observe the Numerator Pattern

The numerators of the sequence terms are as follows: 1, 3, 5, 7, 9. This sequence is an arithmetic sequence with a common difference of 2, starting from 1. Therefore, the numerator of the nth term can be expressed as: \[ 2n - 1 \].

Step 2: Observe the Denominator Pattern

The denominators of the sequence terms are as follows: 1, 4, 9, 16, 25. This sequence is made up of perfect squares: \[ 1^2, 2^2, 3^2, 4^2, 5^2 \].So, the denominator of the nth term is \[ n^2 \].

Step 3: Combine Patterns to Formulate the nth Term

Using the patterns identified in Steps 1 and 2, the nth term of the sequence is given by combining the numerator and denominator formulas. Hence, the nth term is: \[ a_n = \frac{2n - 1}{n^2} \].

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term increases by a constant difference from the previous one. In this exercise, the numerators of the given sequence form an arithmetic sequence: 1, 3, 5, 7, 9. Here, the difference between consecutive terms is 2. This constant difference is known as the 'common difference'.
For any arithmetic sequence, we can find the general formula for the nth term using the formula:

• \[ a_n = a_1 + (n-1) imes d \]

where \( a_1 \) is the first term and \( d \) is the common difference. Applying this formula, the nth term for our numerators becomes:

• \[ a_n = 1 + (n-1) imes 2 = 2n - 1 \]

By recognizing the arithmetic pattern, you can predict any term in the sequence just by knowing the term's position.

Numerator Pattern

In the sequence we're examining, the numerators follow a simple arithmetic pattern. From the term pattern 1, 3, 5, 7, 9, it's clear these are odd numbers starting from 1.
This sequence increases by a regular interval of 2 with each subsequent term. This forms a crucial part of the nth term in our sequence.

• The formula developed for the numerators, based on their arithmetic sequence, is \( 2n - 1 \).

This means for any term position 'n', its numerator can simply be calculated using this formula. Recognizing simple patterns like these can greatly simplify understanding complex sequences.

Denominator Pattern

The denominators of our sequence: 1, 4, 9, 16, 25, show a different kind of pattern known as a sequence of perfect squares. You might notice that each number is simply a square of consecutive natural numbers:

• \( 1^2 = 1 \)
• \( 2^2 = 4 \)
• \( 3^2 = 9 \)
• \( 4^2 = 16 \)
• \( 5^2 = 25 \)

Perfect squares are numbers that result from multiplying a number by itself. So, we can say that the denominator for the nth term is simply \( n^2 \).
Understanding how perfect squares work makes it easier to identify and predict patterns in sequences, directly impacting how you calculate series terms.

Perfect Squares

Perfect squares are numbers like 1, 4, 9, 16, and so on, which are the result of squaring integers. In our sequence, each denominator is a perfect square, expressed as \( n^2\).
This pattern is very useful in sequence calculations because it provides a straightforward way to determine values based on their position in the sequence.

• For example, the denominator of the nth term is expressed neatly using the square function as \( n^2 \).

Recognizing perfect squares helps in reducing the complexity when dealing with sequences, especially when terms are hidden in the squared form. In sequence calculations, these are prevalent and give structure to what might appear as an arbitrary list of numbers.