If the first two consecutive terms of a G.P. are 125 and 25, find its `6^(th)` term.

Answer

To find the sixth term of a Geometric Progression (G.P.) where the first two consecutive terms are given as 125 and 25, we can follow these steps:

### Step 1: Identify the first term (a) and the second term (a * r)
The first term \( a \) is given as 125, and the second term \( a \cdot r \) is given as 25.

### Step 2: Find the common ratio (r)
The common ratio \( r \) can be calculated using the formula:
\[ r = \frac{\text{second term}}{\text{first term}} = \frac{25}{125} \]
Calculating this gives:
\[ r = \frac{1}{5} \]

### Step 3: Use the formula for the nth term of a G.P.
The formula for the nth term \( T_n \) of a G.P. is given by:
\[ T_n = a \cdot r^{(n-1)} \]
We need to find the sixth term, so we set \( n = 6 \):
\[ T_6 = a \cdot r^{(6-1)} = 125 \cdot \left(\frac{1}{5}\right)^{5} \]

### Step 4: Calculate \( \left(\frac{1}{5}\right)^{5} \)
Calculating \( \left(\frac{1}{5}\right)^{5} \):
\[ \left(\frac{1}{5}\right)^{5} = \frac{1}{5^5} = \frac{1}{3125} \]

### Step 5: Substitute back to find \( T_6 \)
Now substituting back into the equation for \( T_6 \):
\[ T_6 = 125 \cdot \frac{1}{3125} \]

### Step 6: Simplify the expression
Calculating this gives:
\[ T_6 = \frac{125}{3125} \]
Since \( 3125 = 125 \cdot 25 \), we can simplify:
\[ T_6 = \frac{1}{25} \]

### Final Answer
Thus, the sixth term of the G.P. is:
\[ \boxed{\frac{1}{25}} \]

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