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). Let's break down the steps to find the sum.

### Step 1: Identify the first term and the common ratio
The first term \(a\) of the series is \(2\). The second term is \(0.2\), which can be expressed as \(2 \times \frac{1}{10}\). The third term is \(0.02\), which can be expressed as \(2 \times \frac{1}{100}\).

From this, we can see that the common ratio \(r\) is:
\[ r = \frac{0.2}{2} = \frac{1}{10} \]

### Step 2: Write the formula for the sum of the first \(n\) terms of a GP
The formula for the sum \(S_n\) of the first \(n\) terms of a geometric series is given by:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

### Step 3: Substitute the values into the formula
Here, we need to find the sum of the first \(7\) terms, so \(n = 7\), \(a = 2\), and \(r = \frac{1}{10}\). Substituting these values into the formula gives:
\[ S_7 = 2 \frac{1 - \left(\frac{1}{10}\right)^7}{1 - \frac{1}{10}} \]

### Step 4: Simplify the expression
First, simplify the denominator:
\[ 1 - \frac{1}{10} = \frac{9}{10} \]

Now substituting this back into the equation:
\[ S_7 = 2 \frac{1 - \left(\frac{1}{10}\right)^7}{\frac{9}{10}} \]

This can be rewritten as:
\[ S_7 = 2 \cdot \frac{10}{9} \left(1 - \left(\frac{1}{10}\right)^7\right) \]

### Step 5: Calculate \(\left(\frac{1}{10}\right)^7\)
Calculating \(\left(\frac{1}{10}\right)^7\):
\[ \left(\frac{1}{10}\right)^7 = \frac{1}{10^7} = \frac{1}{10000000} \]

### Step 6: Substitute back and finalize the sum
Now substituting this value back into the equation:
\[ S_7 = \frac{20}{9} \left(1 - \frac{1}{10000000}\right) \]
This can be simplified to:
\[ S_7 = \frac{20}{9} \left(\frac{9999999}{10000000}\right) \]

### Step 7: Final calculation
Calculating this gives:
\[ S_7 = \frac{20 \times 9999999}{9 \times 10000000} = \frac{199999980}{90000000} \]

Thus, the sum of the first 7 terms of the series is:
\[ S_7 \approx 2.2222 \text{ (to 4 decimal places)} \]