If the `10^(th)` term of an HP is 21 and `21^(st)` term of the same HP is 10, then find the `210^(th)` term.

Answer

To solve the problem, we need to find the 210th term of a Harmonic Progression (HP) given that the 10th term is 21 and the 21st term is 10.

### Step-by-step Solution:

1. **Understanding HP and AP Relationship**:
Since Harmonic Progression (HP) is the reciprocal of an Arithmetic Progression (AP), we can denote the nth term of the HP as \( T_n \) and the corresponding term in the AP as \( A_n \).
\[ T_n = \frac{1}{A_n} \]

2. **Setting Up the Equations**:
Given:
- \( T_{10} = 21 \) implies \( A_{10} = \frac{1}{21} \)
- \( T_{21} = 10 \) implies \( A_{21} = \frac{1}{10} \)

3. **Using the Formula for the nth Term of AP**:
The nth term of an AP is given by:
\[ A_n = a + (n-1) \cdot d \]
where \( a \) is the first term and \( d \) is the common difference.

For \( n = 10 \):
\[ A_{10} = a + 9d = \frac{1}{21} \quad \text{(1)} \]
For \( n = 21 \):
\[ A_{21} = a + 20d = \frac{1}{10} \quad \text{(2)} \]

4. **Subtracting the Equations**:
Subtract equation (1) from equation (2):
\[ (a + 20d) - (a + 9d) = \frac{1}{10} - \frac{1}{21} \]
Simplifying gives:
\[ 11d = \frac{1}{10} - \frac{1}{21} \]

5. **Finding a Common Denominator**:
The common denominator for 10 and 21 is 210:
\[ \frac{1}{10} = \frac{21}{210}, \quad \frac{1}{21} = \frac{10}{210} \]
Thus,
\[ 11d = \frac{21 - 10}{210} = \frac{11}{210} \]

6. **Solving for d**:
Dividing both sides by 11:
\[ d = \frac{1}{210} \]

7. **Finding a**:
Substitute \( d \) back into equation (1):
\[ a + 9 \cdot \frac{1}{210} = \frac{1}{21} \]
This simplifies to:
\[ a + \frac{9}{210} = \frac{10}{210} \]
Therefore:
\[ a = \frac{10}{210} - \frac{9}{210} = \frac{1}{210} \]

8. **Finding the 210th Term**:
Now we can find \( T_{210} \):
\[ A_{210} = a + 209d = \frac{1}{210} + 209 \cdot \frac{1}{210} = \frac{1 + 209}{210} = \frac{210}{210} = 1 \]
Thus,
\[ T_{210} = \frac{1}{A_{210}} = \frac{1}{1} = 1 \]

### Final Answer:
The 210th term of the HP is **1**.