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

A **harmonic sequence** is defined as a sequence of numbers whose reciprocals form an arithmetic sequence. To find the next term of a harmonic sequence, we must first convert the given terms into their reciprocals, examine the arithmetic progression, and then determine the next reciprocal value. After that, we'll find the harmonic term by taking the reciprocal of this value.

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

### Step 1: Find the reciprocals
The reciprocals of these terms are:
\[
-\frac{1}{5}, \frac{7}{15}, \frac{17}{15}, \frac{9}{5}, \frac{37}{15}
\]

### Step 2: Check for the arithmetic sequence
We will now check if the differences between consecutive terms form an arithmetic progression:

1. Difference between \(\frac{7}{15}\) and \(-\frac{1}{5}\):
   \[
   \frac{7}{15} - \left(-\frac{1}{5}\right) = \frac{7}{15} + \frac{1}{5} = \frac{7}{15} + \frac{3}{15} = \frac{10}{15} = \frac{2}{3}
   \]

2. Difference between \(\frac{17}{15}\) and \(\frac{7}{15}\):
   \[
   \frac{17}{15} - \frac{7}{15} = \frac{10}{15} = \frac{2}{3}
   \]

3. Difference between \(\frac{9}{5}\) and \(\frac{17}{15}\):
   \[
   \frac{9}{5} - \frac{17}{15} = \frac{27}{15} - \frac{17}{15} = \frac{10}{15} = \frac{2}{3}
   \]

4. Difference between \(\frac{37}{15}\) and \(\frac{9}{5}\):
   \[
   \frac{37}{15} - \frac{9}{5} = \frac{37}{15} - \frac{27}{15} = \frac{10}{15} = \frac{2}{3}
   \]

### Step 3: Find the next reciprocal term
The differences are constant, which means the reciprocals follow an arithmetic progression with a common difference of \(\frac{2}{3}\). To find the next term, we add \(\frac{2}{3}\) to \(\frac{37}{15}\):

\[
\frac{37}{15} + \frac{2}{3} = \frac{37}{15} + \frac{10}{15} = \frac{47}{15}
\]

### Step 4: Find the next harmonic term
The reciprocal of \(\frac{47}{15}\) is:
\[
\frac{15}{47}
\]

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

---

Would you like a deeper explanation, or have any questions? Here are 5 related questions to explore:

1. How do you identify a harmonic sequence from a set of numbers?
2. What are the key properties of harmonic sequences?
3. How do arithmetic and harmonic sequences differ?
4. What is the relationship between harmonic sequences and harmonic means?
5. Can harmonic sequences include negative terms?

**Tip:** Harmonic sequences are closely related to arithmetic sequences; learning how to transition between the two is key to solving such problems effectively.

Find the next term in the harmonic sequence -5, 15/7, 15/17, 5/9, 15/37. This detailed solution walks through finding the arithmetic progression of reciprocals and determining the next harmonic term. Perfect for Grades 10-12 students studying sequences and series.

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