Suppose (m+n)th term of a G.P. is p and (m-n)th term is q, then its nth is

Options

A. `sqrt( pq)`
B. `p((q)/(p))^(m//n)`
C. `p((q)/(p))^(m//2n)`
D. `sqrt((p)/(q))`

Answer

To solve the problem, we need to find the nth term of a geometric progression (G.P.) given the (m+n)th term is p and the (m-n)th term is q.

### Step-by-Step Solution:

1. **Understand the formula for the nth term of a G.P.**:
The nth term of a G.P. can be expressed as:
\[ T_n = A \cdot r^{n-1} \]
where \( A \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

2. **Write the expressions for the (m+n)th and (m-n)th terms**:
From the problem, we have:
\[ T_{m+n} = A \cdot r^{(m+n)-1} = p \quad \text{(1)} \]
\[ T_{m-n} = A \cdot r^{(m-n)-1} = q \quad \text{(2)} \]

3. **Divide the two equations to eliminate A**:
By dividing equation (1) by equation (2):
\[ \frac{T_{m+n}}{T_{m-n}} = \frac{p}{q} = \frac{A \cdot r^{(m+n)-1}}{A \cdot r^{(m-n)-1}} \]
This simplifies to:
\[ \frac{p}{q} = r^{(m+n-1) - (m-n-1)} = r^{(m+n-1 - m + n + 1)} = r^{2n} \]

4. **Solve for r**:
From the equation above, we can express \( r \) as:
\[ r^{2n} = \frac{p}{q} \implies r = \left(\frac{p}{q}\right)^{\frac{1}{2n}} \]

5. **Substitute r back into one of the original equations to find A**:
We can substitute \( r \) back into equation (1):
\[ p = A \cdot r^{m+n-1} \]
Thus,
\[ A = \frac{p}{r^{m+n-1}} = \frac{p}{\left(\frac{p}{q}\right)^{\frac{(m+n-1)}{2n}}} \]

6. **Simplify A**:
This simplifies to:
\[ A = p \cdot \left(\frac{q}{p}\right)^{\frac{(m+n-1)}{2n}} = p \cdot q^{\frac{(m+n-1)}{2n}} \cdot p^{-\frac{(m+n-1)}{2n}} \]

7. **Find the nth term**:
Now, we can find the nth term:
\[ T_n = A \cdot r^{n-1} \]
Substitute \( A \) and \( r \):
\[ T_n = \left(p \cdot q^{\frac{(m+n-1)}{2n}} \cdot p^{-\frac{(m+n-1)}{2n}}\right) \cdot \left(\frac{p}{q}\right)^{\frac{(n-1)}{2n}} \]

8. **Final expression**:
After simplifying, we arrive at:
\[ T_n = p \cdot \left(\frac{q}{p}\right)^{\frac{m}{2n}} = \frac{p \cdot q^{\frac{m}{2n}}}{p^{\frac{m}{2n}}} \]
Therefore, the nth term is:
\[ T_n = p \cdot \left(\frac{q}{p}\right)^{\frac{m}{2n}} \]