Problem 53 - Precalculus: Mathematical for Calculus

Question

A sequence is harmonic if the reciprocals of the terms of the sequence form an arithmetic sequence. Determine whether the following sequence is harmonic: $$1, \frac{3}{5}, \frac{3}{7}, \frac{1}{3}, \dots$$

Step-by-Step Solution

Answer

The sequence is harmonic.

Step 1: Identify Reciprocals

The given sequence is \(1, \frac{3}{5}, \frac{3}{7}, \frac{1}{3}, \dots\). To determine if this sequence is harmonic, we first find the reciprocals of each term. The reciprocals are: \[1, \frac{5}{3}, \frac{7}{3}, 3, \dots\]

Step 2: Check Arithmetic Progression of Reciprocals

Now, we need to verify if the sequence of reciprocals \(1, \frac{5}{3}, \frac{7}{3}, 3, \dots\) forms an arithmetic sequence. Recall that a sequence is arithmetic if the difference between consecutive terms is constant.

Step 3: Calculate Common Difference

Find the differences between each pair of consecutive terms in the reciprocals sequence: - \(\frac{5}{3} - 1 = \frac{5}{3} - \frac{3}{3} = \frac{2}{3}\)- \(\frac{7}{3} - \frac{5}{3} = \frac{2}{3}\)- \(3 - \frac{7}{3} = \frac{9}{3} - \frac{7}{3} = \frac{2}{3}\)The difference is constant and equal to \(\frac{2}{3}\).

Step 4: Conclusion

Since the sequence of reciprocals forms an arithmetic sequence with a common difference of \(\frac{2}{3}\), the original sequence \(1, \frac{3}{5}, \frac{3}{7}, \frac{1}{3}, \dots\) is harmonic.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant is known as the "common difference". For example, in the sequence 2, 4, 6, 8, the difference between each consecutive number is 2. This same principle applies to any arithmetic sequence, regardless of whether the numbers are increasing or decreasing.

To check if a series of numbers is arithmetic, you simply subtract the previous term from the current term and see if this result is the same throughout the series. For instance, in the sequence 3, 7, 11, 15:

• 7 - 3 = 4
• 11 - 7 = 4
• 15 - 11 = 4

Since the common difference is constant at 4, it confirms an arithmetic sequence. Arithmetic sequences are foundational to understanding various mathematical concepts, including harmonic sequences, which involve reciprocal relationships.

Reciprocals

In mathematics, a reciprocal of a number is simply one divided by that number. So, if you have a number 'a', its reciprocal is defined as \( \frac{1}{a} \). Some easy-to-understand examples include:

• The reciprocal of 2 is \( \frac{1}{2} \).
• The reciprocal of 5 is \( \frac{1}{5} \).
• The reciprocal of \( \frac{1}{3} \) is 3, since \( \frac{1}{\left(\frac{1}{3}\right)} = 3 \).

Reciprocals are fundamental for understanding harmonic sequences, as these sequences require that the reciprocals of their terms create an arithmetic sequence. Therefore, by converting each term of the sequence into its reciprocal and checking whether these reciprocals form a consistent arithmetic pattern, we can determine if the original sequence is harmonic.

In the given problem, transforming the sequence \( 1, \frac{3}{5}, \frac{3}{7}, \frac{1}{3} \) into its reciprocals was the first step to identify whether it can form an arithmetic sequence.

Common Difference

The common difference in an arithmetic sequence is the amount added (or sometimes subtracted) from one term to the next. It is denoted by 'd'. When establishing if a sequence is arithmetic, as we've done with the reciprocals, calculating the common difference accurately is crucial.

• In an arithmetic sequence, calculate by subtracting the first term from the second, the second from the third, and so forth, to find a repeating difference.
• For our sequence \( 1, \frac{3}{5}, \frac{3}{7}, \frac{1}{3} \), after taking reciprocals, we assess \( 1, \frac{5}{3}, \frac{7}{3}, 3 \).
• From the calculations: \( \frac{5}{3} - 1 = \frac{2}{3} \), \( \frac{7}{3} - \frac{5}{3} = \frac{2}{3} \), \( 3 - \frac{7}{3} = \frac{2}{3} \).

Each pair of consecutive terms has the same difference of \( \frac{2}{3} \), illustrating that the reciprocals form an arithmetic sequence. Knowing the common difference is essential, as it confirms the arithmetic nature of the sequence of reciprocals, thus proving the original sequence is harmonic.