Find the sum of first n terms of:
`0.7+0.77+0.777+`…..

Answer

To find the sum of the first n terms of the series \(0.7 + 0.77 + 0.777 + \ldots\), we can follow these steps:

### Step 1: Rewrite the series
We can express the terms in a different form:
\[ 0.7 = \frac{7}{10}, \quad 0.77 = \frac{77}{100} = \frac{7 \times 11}{100}, \quad 0.777 = \frac{777}{1000} = \frac{7 \times 111}{1000} \]
Thus, the series can be rewritten as:
\[ S_n = 0.7 + 0.77 + 0.777 + \ldots + \text{(n terms)} \]

### Step 2: Factor out the common term
Notice that each term can be expressed as:
\[ S_n = 7 \left(\frac{1}{10} + \frac{11}{100} + \frac{111}{1000} + \ldots\right) \]
This can be simplified to:
\[ S_n = 7 \left(0.1 + 0.11 + 0.111 + \ldots\right) \]

### Step 3: Express the series in terms of a geometric series
We can express \(0.1 + 0.11 + 0.111 + \ldots\) as:
\[ 0.1 + 0.11 + 0.111 + \ldots = 0.1(1 + 1 + 1 + \ldots) + (0.01 + 0.001 + \ldots) \]
The first part \(0.1(1 + 1 + 1 + \ldots)\) is simply \(0.1n\) (since there are n terms).

### Step 4: Sum the geometric series
The second part \(0.01 + 0.001 + \ldots\) is a geometric series with the first term \(a = 0.1\) and common ratio \(r = 0.1\):
\[ \text{Sum} = \frac{a(1 - r^n)}{1 - r} = \frac{0.1(1 - (0.1)^n)}{1 - 0.1} = \frac{0.1(1 - 0.1^n)}{0.9} \]
Thus, we can write:
\[ S_n = 7 \left(n \cdot 0.1 - \frac{0.1(1 - 0.1^n)}{0.9}\right) \]

### Step 5: Simplify the expression
Combining the terms, we have:
\[ S_n = \frac{7}{9} \left( n - \frac{1 - 0.1^n}{10} \right) \]
This can be further simplified to:
\[ S_n = \frac{7}{9} \left( 9n - 1 + 0.1^n \right) \]

### Final Answer
Thus, the sum of the first n terms is:
\[ S_n = \frac{7}{81} (9n - 1 + 10^{-n}) \]