The sum of series `(7)/(11)+(77)/(11^(2))+(777)/(11^(3))+(7777)/(11^(4))+……oo` is

Options

A. `(69)/(77)`
B. `(77)/(11)`
C. `(70)/(99)`
D. `(77)/(10)`

Answer

To find the sum of the series

\[ S = \frac{7}{11} + \frac{77}{11^2} + \frac{777}{11^3} + \frac{7777}{11^4} + \ldots \]

we can first rewrite the terms in a more manageable form.

### Step 1: Rewrite the series

Notice that each term in the series can be expressed as follows:

- The first term is \( \frac{7}{11} = \frac{7 \cdot 1}{11^1} \)
- The second term is \( \frac{77}{11^2} = \frac{7 \cdot 11}{11^2} \)
- The third term is \( \frac{777}{11^3} = \frac{7 \cdot 111}{11^3} \)
- The fourth term is \( \frac{7777}{11^4} = \frac{7 \cdot 1111}{11^4} \)

We can see that the numerators can be expressed as \( 7 \cdot (1 + 10 + 100 + \ldots) \).

### Step 2: Factor out the common term

We can factor out \( \frac{7}{11} \):

\[ S = \frac{7}{11} \left( 1 + \frac{11}{11} + \frac{111}{11^2} + \frac{1111}{11^3} + \ldots \right) \]

### Step 3: Express the series in a simpler form

Now, let's denote the inner series as \( T \):

\[ T = 1 + \frac{11}{11} + \frac{111}{11^2} + \frac{1111}{11^3} + \ldots \]

### Step 4: Multiply the series by \( \frac{1}{11} \)

Now, multiply \( T \) by \( \frac{1}{11} \):

\[ \frac{T}{11} = \frac{1}{11} + \frac{11}{11^2} + \frac{111}{11^3} + \frac{1111}{11^4} + \ldots \]

### Step 5: Subtract the two equations

Now, subtract \( \frac{T}{11} \) from \( T \):

\[ T - \frac{T}{11} = 1 + \left( \frac{11}{11} - \frac{1}{11} \right) + \left( \frac{111}{11^2} - \frac{11}{11^2} \right) + \left( \frac{1111}{11^3} - \frac{111}{11^3} \right) + \ldots \]

This simplifies to:

\[ \frac{10T}{11} = 1 + \frac{10}{11} + \frac{100}{11^2} + \frac{1000}{11^3} + \ldots \]

### Step 6: Recognize the geometric series

The series on the right is a geometric series with first term \( 1 \) and common ratio \( \frac{10}{11} \):

\[ \text{Sum} = \frac{1}{1 - \frac{10}{11}} = 11 \]

### Step 7: Solve for \( T \)

Now we have:

\[ \frac{10T}{11} = 11 \]

Multiplying both sides by \( \frac{11}{10} \):

\[ T = \frac{11 \cdot 11}{10} = \frac{121}{10} \]

### Step 8: Substitute back to find \( S \)

Now substitute \( T \) back into the equation for \( S \):

\[ S = \frac{7}{11} \cdot T = \frac{7}{11} \cdot \frac{121}{10} = \frac{847}{110} \]

### Final Result

Thus, the sum of the series is:

\[ S = \frac{77}{10} \]