Find the `8^(th)` term of the geometric progression : 5,10 20, . . . . . . . . .

Answer

To find the 8th term of the geometric progression (GP) given by the sequence: 5, 10, 20, ..., we can follow these steps:

### Step-by-Step Solution:

1. **Identify the first term (a)**:
The first term of the GP is \( a = 5 \).

2. **Determine the common ratio (r)**:
The common ratio \( r \) can be found by dividing the second term by the first term or the third term by the second term.
\[ r = \frac{t_2}{t_1} = \frac{10}{5} = 2 \]
We can also verify this with the third term:
\[ r = \frac{t_3}{t_2} = \frac{20}{10} = 2 \]
Thus, the common ratio \( r = 2 \).

3. **Use the formula for the nth term of a GP**:
The formula for the nth term \( t_n \) of a geometric progression is given by:
\[ t_n = a \cdot r^{(n-1)} \]
For the 8th term \( t_8 \):
\[ t_8 = a \cdot r^{(8-1)} = 5 \cdot 2^{7} \]

4. **Calculate \( 2^7 \)**:
\[ 2^7 = 128 \]

5. **Calculate \( t_8 \)**:
\[ t_8 = 5 \cdot 128 = 640 \]

### Final Answer:
The 8th term of the geometric progression is \( t_8 = 640 \).

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