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Solve this question Find the indicated term of the following harmonic sequences.

To solve the given problem, we need to find the specified terms of harmonic sequences. The harmonic sequence is given in the form of fractions. A harmonic sequence is the reciprocal of an arithmetic sequence.

### Let’s break down each sequence and find the specified terms.

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#### 1. Sequence 1:  
\[ \frac{1}{4}, \frac{1}{8}, \frac{1}{12}, \dots \]  
We need to find the 9th term.

- The corresponding arithmetic sequence for this harmonic sequence is:  
\[ 4, 8, 12, \dots \]

- The common difference is:  
\[ d = 8 - 4 = 4 \]

- The nth term of an arithmetic sequence is given by the formula:  
\[ a_n = a_1 + (n-1)d \]  
Where \(a_1 = 4\) and \(d = 4\).

- For the 9th term:  
\[ a_9 = 4 + (9-1) \times 4 = 4 + 32 = 36 \]

Thus, the 9th term of the harmonic sequence is:  
\[ \frac{1}{36} \]

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#### 2. Sequence 2:  
\[ \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \dots \]  
We need to find the 10th term.

- The corresponding arithmetic sequence is:  
\[ 3, 5, 7, \dots \]

- The common difference is:  
\[ d = 5 - 3 = 2 \]

- Using the arithmetic sequence formula for the nth term:  
\[ a_n = a_1 + (n-1)d \]  
Where \(a_1 = 3\) and \(d = 2\).

- For the 10th term:  
\[ a_{10} = 3 + (10-1) \times 2 = 3 + 18 = 21 \]

Thus, the 10th term of the harmonic sequence is:  
\[ \frac{1}{21} \]

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#### 3. Sequence 3:  
\[ \frac{1}{5}, \frac{1}{7}, \frac{1}{9}, \dots \]  
We need to find the 7th term.

- The corresponding arithmetic sequence is:  
\[ 5, 7, 9, \dots \]

- The common difference is:  
\[ d = 7 - 5 = 2 \]

- Using the arithmetic sequence formula for the nth term:  
\[ a_n = a_1 + (n-1)d \]  
Where \(a_1 = 5\) and \(d = 2\).

- For the 7th term:  
\[ a_7 = 5 + (7-1) \times 2 = 5 + 12 = 17 \]

Thus, the 7th term of the harmonic sequence is:  
\[ \frac{1}{17} \]

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### Summary of Results:
1. 9th term of the first sequence: \( \frac{1}{36} \)
2. 10th term of the second sequence: \( \frac{1}{21} \)
3. 7th term of the third sequence: \( \frac{1}{17} \)

Would you like more detailed explanations for any of these steps?

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Here are 5 related questions for further practice:
1. What is the 12th term of the harmonic sequence \( \frac{1}{6}, \frac{1}{9}, \frac{1}{12}, \dots \)?
2. How do you derive the formula for the nth term of an arithmetic sequence?
3. Can you calculate the sum of the first 5 terms of the harmonic sequence \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots \)?
4. What is the 15th term of the harmonic sequence with first term \( \frac{1}{7} \) and second term \( \frac{1}{14} \)?
5. How does the relationship between harmonic and arithmetic sequences help in solving harmonic sequences?

**Tip:** When working with harmonic sequences, always convert them to arithmetic sequences by taking the reciprocal, as it simplifies finding the nth term!

Find the indicated term of the following harmonic sequences.

Solve harmonic sequence problems to find the 9th, 10th, and 7th terms. Learn about harmonic and arithmetic sequences and apply the nth term formula to calculate precise values. Suitable for students in Grades 9-10.

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