In harmonic sequence, What is the next term of -5, 15/7, 15/17, 5/9, 15/37?

To find the next term in a harmonic sequence, we first need to understand that the terms of a harmonic sequence are the reciprocals of an arithmetic sequence. So, let's begin by converting the given harmonic sequence into its corresponding arithmetic sequence by taking the reciprocal of each term.

The harmonic sequence given is:
\[
-5, \frac{15}{7}, \frac{15}{17}, \frac{5}{9}, \frac{15}{37}
\]

The reciprocals (to form the arithmetic sequence) are:
\[
-\frac{1}{5}, \frac{7}{15}, \frac{17}{15}, \frac{9}{5}, \frac{37}{15}
\]

Next, let's check if these terms form an arithmetic sequence by finding the common difference between consecutive terms:

1. \(\frac{7}{15} - \left(-\frac{1}{5}\right) = \frac{7}{15} + \frac{3}{15} = \frac{10}{15} = \frac{2}{3}\)
2. \(\frac{17}{15} - \frac{7}{15} = \frac{10}{15} = \frac{2}{3}\)
3. \(\frac{9}{5} - \frac{17}{15} = \frac{27}{15} - \frac{17}{15} = \frac{10}{15} = \frac{2}{3}\)
4. \(\frac{37}{15} - \frac{9}{5} = \frac{37}{15} - \frac{27}{15} = \frac{10}{15} = \frac{2}{3}\)

As we can see, the common difference between consecutive terms in the arithmetic sequence is \(\frac{2}{3}\).

Now, to find the next term in the arithmetic sequence, we add \(\frac{2}{3}\) to the last term:
\[
\frac{37}{15} + \frac{2}{3} = \frac{37}{15} + \frac{10}{15} = \frac{47}{15}
\]

The reciprocal of \(\frac{47}{15}\) will give us the next term in the harmonic sequence:
\[
\frac{15}{47}
\]

Thus, the next term in the harmonic sequence is \(\frac{15}{47}\).

Would you like further details on harmonic sequences or arithmetic sequences?

Here are five related questions to consider:
1. How can harmonic sequences be applied in real-life problems?
2. Can you derive the formula for the \(n\)-th term of a harmonic sequence?
3. What is the difference between harmonic and geometric sequences?
4. How can you solve problems involving the sum of a harmonic series?
5. How do arithmetic progressions relate to other types of sequences?

**Tip**: Always convert a harmonic sequence into its arithmetic counterpart to simplify calculations and problem-solving.

In harmonic sequence, what is the next term of -5, 15/7, 15/17, 5/9, 15/37?

Discover how to find the next term in a harmonic sequence using reciprocals of an arithmetic sequence. This step-by-step solution covers essential concepts like harmonic sequences, arithmetic sequences, and the common difference. Perfect for high school students in Grades 9-12.

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