Find the sum to indicated number of terms in each of the geometric progressions in Questions 7 to 10 :
0.15, 0.015, 0.0015,…20 terms.

Explanation

To find the sum of the first 20 terms of the given geometric progression (GP) 0.15, 0.015, 0.0015, we will follow these steps:

### Step 1: Identify the first term (A) and the common ratio (R)
The first term \( A \) of the GP is:
\[ A = 0.15 \]

To find the common ratio \( R \), we can divide the second term by the first term:
\[ R = \frac{0.015}{0.15} = \frac{15}{150} = \frac{1}{10} = 0.1 \]

### Step 2: Use the formula for the sum of the first n terms of a GP
The formula for the sum of the first \( n \) terms of a geometric progression is given by:
\[ S_n = A \frac{1 - R^n}{1 - R} \]
where \( n \) is the number of terms.

### Step 3: Substitute the values into the formula
We need to find \( S_{20} \):
\[ S_{20} = 0.15 \frac{1 - (0.1)^{20}}{1 - 0.1} \]

### Step 4: Calculate \( 1 - R \)
Calculating \( 1 - R \):
\[ 1 - 0.1 = 0.9 \]

### Step 5: Calculate \( (0.1)^{20} \)
Calculating \( (0.1)^{20} \):
\[ (0.1)^{20} = 10^{-20} \]

### Step 6: Substitute back into the sum formula
Now substituting back into the formula:
\[ S_{20} = 0.15 \frac{1 - 10^{-20}}{0.9} \]

### Step 7: Simplify the expression
\[ S_{20} = 0.15 \cdot \frac{1 - 10^{-20}}{0.9} \]
Calculating the fraction:
\[ S_{20} = \frac{0.15}{0.9} (1 - 10^{-20}) = \frac{1}{6} (1 - 10^{-20}) \]

### Step 8: Final calculation
Thus, the final sum can be expressed as:
\[ S_{20} = \frac{1}{6} (1 - 10^{-20}) \]