What is the formula for harmonic sequence?
Give (1 only) sample problem with solution

A **harmonic sequence** is a sequence of numbers where the reciprocals of the terms form an arithmetic sequence. The formula for the \(n\)-th term of a harmonic sequence can be derived from the corresponding arithmetic sequence.

### Formula:
Given a harmonic sequence, if the reciprocals of its terms form an arithmetic sequence, then the \(n\)-th term of the harmonic sequence \(H_n\) can be expressed as:
\[
H_n = \frac{1}{A_1 + (n-1) \cdot d}
\]
where:
- \(A_1\) is the first term of the arithmetic sequence of the reciprocals,
- \(d\) is the common difference of the arithmetic sequence.

### Sample Problem:
**Problem:**  
Find the 5th term of a harmonic sequence where the first term is \(H_1 = 1\) and the second term is \(H_2 = \frac{1}{2}\).

**Solution:**
1. First, find the first two terms of the corresponding arithmetic sequence of reciprocals:
   \[
   \text{Let the reciprocals of } H_1 \text{ and } H_2 \text{ be } A_1 \text{ and } A_2.
   \]
   \[
   A_1 = \frac{1}{H_1} = 1 \quad \text{and} \quad A_2 = \frac{1}{H_2} = 2
   \]
   
2. Calculate the common difference \(d\) of the arithmetic sequence:
   \[
   d = A_2 - A_1 = 2 - 1 = 1
   \]

3. Use the formula for the \(n\)-th term of the arithmetic sequence of reciprocals to find the 5th term \(A_5\):
   \[
   A_5 = A_1 + (5-1) \cdot d = 1 + 4 \cdot 1 = 5
   \]

4. The 5th term \(H_5\) of the harmonic sequence is the reciprocal of \(A_5\):
   \[
   H_5 = \frac{1}{A_5} = \frac{1}{5}
   \]

Thus, the 5th term of the harmonic sequence is \(\boxed{\frac{1}{5}}\).

Would you like more details or have any questions?

Here are five related questions you can explore:

1. How do you derive the formula for the \(n\)-th term of a harmonic sequence?
2. What are the properties of harmonic sequences?
3. How do harmonic sequences relate to harmonic means?
4. Can you find the sum of the first \(n\) terms of a harmonic sequence?
5. How does the harmonic sequence differ from geometric and arithmetic sequences?

**Tip:** In harmonic sequences, understanding the corresponding arithmetic sequence of the reciprocals is key to solving problems efficiently.

What is the formula for harmonic sequence? Give (1 only) sample problem with solution

Explore the formula for the n-th term of a harmonic sequence and solve a sample problem. This page covers sequences, arithmetic sequences, and the properties of harmonic sequences. Ideal for students in Grades 10-12.

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