Find the 9th term of the harmonic sequence -5, 15/7, 15/17, 5/9, 15/37.

A harmonic sequence is defined by the reciprocals of the terms of an arithmetic sequence. To find the 9th term of the harmonic sequence, we need to identify the underlying arithmetic sequence.

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

### Step 1: Convert to reciprocals (arithmetic sequence)
Take the reciprocals of each term in the harmonic sequence to form the corresponding arithmetic sequence:

- Reciprocal of \(-5\): \(-\frac{1}{5}\)
- Reciprocal of \(\frac{15}{7}\): \(\frac{7}{15}\)
- Reciprocal of \(\frac{15}{17}\): \(\frac{17}{15}\)
- Reciprocal of \(\frac{5}{9}\): \(\frac{9}{5}\)
- Reciprocal of \(\frac{15}{37}\): \(\frac{37}{15}\)

So, the arithmetic sequence is:  
\(-\frac{1}{5}, \frac{7}{15}, \frac{17}{15}, \frac{9}{5}, \frac{37}{15}\)

### Step 2: Find the common difference
To find the common difference \(d\), subtract consecutive terms:

- \(d_1 = \frac{7}{15} - \left(-\frac{1}{5}\right) = \frac{7}{15} + \frac{3}{15} = \frac{10}{15} = \frac{2}{3}\)
- \(d_2 = \frac{17}{15} - \frac{7}{15} = \frac{10}{15} = \frac{2}{3}\)
- \(d_3 = \frac{9}{5} - \frac{17}{15} = \frac{27}{15} - \frac{17}{15} = \frac{10}{15} = \frac{2}{3}\)
- \(d_4 = \frac{37}{15} - \frac{9}{5} = \frac{37}{15} - \frac{27}{15} = \frac{10}{15} = \frac{2}{3}\)

So, the common difference \(d = \frac{2}{3}\).

### Step 3: Find the 9th term of the arithmetic sequence
The general term of an arithmetic sequence is given by:  
\[ a_n = a_1 + (n - 1) d \]

Where \(a_1 = -\frac{1}{5}\) and \(d = \frac{2}{3}\), and we want the 9th term (\(n = 9\)).

\[
a_9 = -\frac{1}{5} + (9 - 1) \times \frac{2}{3}
\]
\[
a_9 = -\frac{1}{5} + 8 \times \frac{2}{3} = -\frac{1}{5} + \frac{16}{3}
\]

To add these fractions, we need a common denominator:

\[
-\frac{1}{5} = -\frac{3}{15}, \quad \frac{16}{3} = \frac{80}{15}
\]
\[
a_9 = -\frac{3}{15} + \frac{80}{15} = \frac{77}{15}
\]

### Step 4: Find the 9th term of the harmonic sequence
Now, take the reciprocal of \(\frac{77}{15}\):

\[
h_9 = \frac{15}{77}
\]

Thus, the 9th term of the harmonic sequence is \(\frac{15}{77}\).

Would you like further details or have any questions?

Here are five related questions to explore further:
1. How do you find the common difference in an arithmetic sequence?
2. What is the formula for the \(n\)th term of an arithmetic sequence?
3. How is the harmonic mean related to harmonic sequences?
4. How do you calculate the sum of the first \(n\) terms in an arithmetic sequence?
5. How do you convert a harmonic sequence into an arithmetic sequence?

**Tip**: The reciprocal of the average of numbers in an arithmetic sequence can give the harmonic mean.

Learn how to find the 9th term of the harmonic sequence -5, 15/7, 15/17, 5/9, and 15/37 by converting it into an arithmetic sequence. Understand the use of reciprocals and the formula for the general term of an arithmetic sequence. Suitable for high school students in Grades 10-12.

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