The first term and the last term of a GP are a and k respectively. If the number of terms be n, then n is equal to (`r to` common ratio),

Options

A. `1-(log k- loga)/(log r)`
B. `1+ (log a+ log k)/(log r)`
C. `1+ (log k - log a)/(log r)`
D. log r = log k - log a

Answer

To solve the problem, we need to find the number of terms \( n \) in a geometric progression (GP) where the first term is \( a \) and the last term is \( k \), with \( r \) being the common ratio.

### Step-by-Step Solution:

1. **Understanding the General Term of a GP:**
The general term \( T_n \) of a geometric progression 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 number of terms.

**Hint:** Remember that the last term can be expressed using the first term and the common ratio.

2. **Setting Up the Equation for the Last Term:**
Since the last term of the GP is given as \( k \), we can write:
\[ k = a \cdot r^{n-1} \]

**Hint:** This equation relates the first term, the common ratio, and the number of terms to the last term.

3. **Taking Logarithms:**
To solve for \( n \), we take the logarithm of both sides:
\[ \log(k) = \log(a \cdot r^{n-1}) \]

**Hint:** Using logarithmic properties will help simplify the equation.

4. **Applying Logarithmic Properties:**
We can use the property of logarithms that states \( \log(a \cdot b) = \log(a) + \log(b) \):
\[ \log(k) = \log(a) + \log(r^{n-1}) \]
Now, using the property \( \log(b^c) = c \cdot \log(b) \), we rewrite:
\[ \log(k) = \log(a) + (n-1) \cdot \log(r) \]

**Hint:** Break down the logarithm of the product into a sum of logarithms.

5. **Rearranging the Equation:**
Rearranging the equation gives:
\[ \log(k) - \log(a) = (n-1) \cdot \log(r) \]

**Hint:** Isolate \( n-1 \) to prepare for solving for \( n \).

6. **Solving for \( n \):**
Dividing both sides by \( \log(r) \):
\[ n - 1 = \frac{\log(k) - \log(a)}{\log(r)} \]
Adding 1 to both sides:
\[ n = 1 + \frac{\log(k) - \log(a)}{\log(r)} \]

**Hint:** This final expression gives you the number of terms in the GP.

### Final Answer:
Thus, the number of terms \( n \) in the geometric progression is:
\[ n = 1 + \frac{\log(k) - \log(a)}{\log(r)} \]