Find the sum of 7 terms of the series 2+0.2+0.02+….

Explanation

To find the sum of the first 7 terms of the series \(2 + 0.2 + 0.02 + \ldots\), we can observe that this series is a geometric progression (GP).

### Step-by-Step Solution:

1. **Identify the first term (a) and the common ratio (r)**:
- The first term \(a = 2\).
- The second term is \(0.2\). To find the common ratio \(r\), we divide the second term by the first term:
\[ r = \frac{0.2}{2} = 0.1 \]

2. **Check if the series is a GP**:
- Since the common ratio \(r = 0.1\) is less than 1, we can use the formula for the sum of the first \(n\) terms of a geometric series.

3. **Use the formula for the sum of the first \(n\) terms of a GP**:
- The formula for the sum of the first \(n\) terms \(S_n\) of a geometric series is:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
- Here, we need to find \(S_7\) (the sum of the first 7 terms), where \(n = 7\).

4. **Substitute the values into the formula**:
- Substitute \(a = 2\), \(r = 0.1\), and \(n = 7\):
\[ S_7 = 2 \frac{1 - (0.1)^7}{1 - 0.1} \]

5. **Calculate \(1 - (0.1)^7\)**:
- Calculate \(0.1^7 = \frac{1}{10^7} = \frac{1}{10000000}\).
- Therefore, \(1 - (0.1)^7 = 1 - \frac{1}{10000000} = \frac{9999999}{10000000}\).

6. **Calculate \(1 - 0.1\)**:
- \(1 - 0.1 = 0.9\).

7. **Substitute back into the formula**:
- Now substituting back, we have:
\[ S_7 = 2 \frac{\frac{9999999}{10000000}}{0.9} \]

8. **Simplify the expression**:
- This simplifies to:
\[ S_7 = 2 \cdot \frac{9999999}{10000000 \cdot 0.9} = 2 \cdot \frac{9999999}{9000000} \]
- Simplifying further:
\[ S_7 = \frac{19999998}{9000000} \]
- This can be simplified to:
\[ S_7 = \frac{20}{9} \left(1 - \frac{1}{10000000}\right) \]

9. **Final result**:
- Therefore, the sum of the first 7 terms of the series is:
\[ S_7 \approx \frac{20}{9} \text{ (approximately 2.222)} \]