Problem 27 - Precalculus: Mathematical for Calculus

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 \( \frac{2n - 1}{n^2} \).

Step 1: Observe the Numerator Pattern

Look at the sequence of numerators: 1, 3, 5, 7, 9, ... . This sequence follows an arithmetic pattern where each number increases by 2. This is the sequence of odd numbers. Thus, the numerator of the nth term can be expressed as the nth odd number.

Step 2: Express the Numerator Formulaically

The nth odd number can be expressed by the formula for odd numbers: \( a_n = 2n - 1 \). Therefore, the formula for the numerator of the nth term is \( 2n - 1 \).

Step 3: Observe the Denominator Pattern

Now examine the denominators: 1, 4, 9, 16, 25, ... . This sequence is composed of perfect squares: \( 1^2, 2^2, 3^2, 4^2, 5^2, ... \). Thus, the nth term's denominator follows the pattern \( n^2 \).

Step 4: Express the Denominator Formulaically

Based on the above observation, the nth term's denominator is \( n^2 \).

Step 5: Combine Numerator and Denominator

Combine the findings from previous steps to express the nth term of the sequence as \( \frac{2n - 1}{n^2} \).

Step 6: Conclusion

The general formula for the nth term is \( a_n = \frac{2n - 1}{n^2} \). This formula allows us to find any term in the given sequence.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. This difference is known as the "common difference." In our context, the sequence of numerators in the fraction sequence provided is an arithmetic sequence. Here's how it works:

• Numbers: 1, 3, 5, 7, 9, ...
• Common difference: 2 (since each term increases by 2)

This sequence of numerators follows a predictable pattern. No matter which term you look at in the sequence, you can always find the next term by adding the common difference to it. This is why we can say that the numerators belong to an arithmetic sequence with a common difference of 2.

Odd Numbers

Odd numbers are integers that cannot be divided evenly by 2. In mathematics, they follow a straightforward sequence: 1, 3, 5, 7, 9, and so on. These numbers can be represented by the formula:

• Formula: \(a_n = 2n - 1\)

This formula simply plugs in the position number (n) to create any odd number in the sequence. For example, when \(n = 1\), \(a_1 = 2 \cdot 1 - 1 = 1\), and when \(n = 2\), \(a_2 = 2 \cdot 2 - 1 = 3\). This pattern accurately describes the sequence of numerators found in the original exercise. Recognizing odd numbers helps in forming a relationship between sequence terms and their positions.

Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer with itself. This means they follow the sequence of squaring natural numbers, such as 1, 4, 9, 16, 25, etc. Each perfect square can be represented by:

• Formula: \(n^2\)

In the given sequence, the denominators are these perfect squares. As you examine each fraction, the denominator increases in the form of increasing perfect squares. Let's see how this applies:

• For \(n = 1\), \(1^2 = 1\)
• For \(n = 2\), \(2^2 = 4\)
• For \(n = 3\), \(3^2 = 9\)

Understanding perfect squares allows us to quickly identify and calculate terms within sequences like this. The unique quality of perfect squares gives every term a consistent mathematical foundation, helping to understand the overall pattern of the sequence.