(i) Find the 10 th term of the G.P.
5,25,125……………..
(ii) Find the nth term of the G.P. 5,25,125,……

Answer

To solve the given problem step by step, we will break it down into two parts as per the question.

### Part (i): Find the 10th term of the G.P. 5, 25, 125...

**Step 1: Identify the first term and common ratio.**
- The first term \( a_1 \) of the G.P. is \( 5 \).
- The common ratio \( r \) can be calculated as:
\[ r = \frac{a_2}{a_1} = \frac{25}{5} = 5 \]

**Step 2: Use the formula for the nth term of a G.P.**
- The formula for the nth term \( a_n \) of a G.P. is given by:
\[ a_n = a_1 \cdot r^{n-1} \]
- Here, we need to find the 10th term \( a_{10} \).

**Step 3: Substitute the values into the formula.**
- Substitute \( a_1 = 5 \), \( r = 5 \), and \( n = 10 \):
\[ a_{10} = 5 \cdot 5^{10-1} = 5 \cdot 5^9 \]

**Step 4: Simplify the expression.**
- Using the property of exponents:
\[ a_{10} = 5^{1 + 9} = 5^{10} \]

**Step 5: Calculate the value.**
- \( 5^{10} = 9765625 \)

Thus, the 10th term of the G.P. is **9765625**.

### Part (ii): Find the nth term of the G.P. 5, 25, 125...

**Step 1: Use the same values for the first term and common ratio.**
- As established earlier, \( a_1 = 5 \) and \( r = 5 \).

**Step 2: Write the formula for the nth term.**
- We will use the same formula for the nth term:
\[ a_n = a_1 \cdot r^{n-1} \]

**Step 3: Substitute the values into the formula.**
- Substitute \( a_1 = 5 \) and \( r = 5 \):
\[ a_n = 5 \cdot 5^{n-1} \]

**Step 4: Simplify the expression.**
- Using the property of exponents:
\[ a_n = 5^1 \cdot 5^{n-1} = 5^{1 + (n-1)} = 5^n \]

Thus, the nth term of the G.P. is **\( 5^n \)**.

### Summary of Answers:
- (i) The 10th term of the G.P. is **9765625**.
- (ii) The nth term of the G.P. is **\( 5^n \)**.