The sum of the infinite series `(5)/(17)+(55)/((17)^(2))+(555)/((17)^(3))+…..` is

Options

A. `(1)/(17)`
B. `(85)/(112)`
C. `(85)/(324)`
D. `(19)/(289)`

Answer

To find the sum of the infinite series

\[ S = \frac{5}{17} + \frac{55}{17^2} + \frac{555}{17^3} + \ldots \]

we can observe that the terms in the series can be expressed in a more manageable form.

### Step 1: Identify the pattern in the series
The terms can be rewritten as:
- First term: \( \frac{5}{17} = \frac{5 \cdot 1}{17^1} \)
- Second term: \( \frac{55}{17^2} = \frac{5 \cdot 11}{17^2} \)
- Third term: \( \frac{555}{17^3} = \frac{5 \cdot 111}{17^3} \)

We can see that each term can be represented as \( \frac{5 \cdot (10^n - 1)/9}{17^{n+1}} \) for \( n = 0, 1, 2, \ldots \).

### Step 2: Rewrite the series
Thus, we can express the series as:
\[ S = 5 \left( \frac{1}{17} + \frac{11}{17^2} + \frac{111}{17^3} + \ldots \right) \]

### Step 3: Factor out the common term
Notice that:
\[ \frac{11}{17^2} = \frac{10 + 1}{17^2} = \frac{10}{17^2} + \frac{1}{17^2} \]
\[ \frac{111}{17^3} = \frac{100 + 10 + 1}{17^3} = \frac{100}{17^3} + \frac{10}{17^3} + \frac{1}{17^3} \]

This suggests that the series can be broken down into a geometric series.

### Step 4: Recognize the geometric series
The series can be expressed as:
\[ S = 5 \left( \frac{1}{17} + \frac{10}{17^2} + \frac{100}{17^3} + \ldots \right) \]

This is a geometric series where the first term \( a = \frac{1}{17} \) and the common ratio \( r = \frac{10}{17} \).

### Step 5: Sum the geometric series
The sum of an infinite geometric series is given by the formula:
\[ S = \frac{a}{1 - r} \]
Substituting in our values:
\[ S = \frac{\frac{1}{17}}{1 - \frac{10}{17}} = \frac{\frac{1}{17}}{\frac{7}{17}} = \frac{1}{7} \]

### Step 6: Multiply by the factor we factored out
Now, we multiply this sum by 5:
\[ S = 5 \cdot \frac{1}{7} = \frac{5}{7} \]

### Final Answer
Thus, the sum of the infinite series is:
\[ \boxed{\frac{5}{7}} \]