Find the sumiof 50 terms of a sequence `7,7.7,7.77,7.777`.

Answer

To find the sum of the first 50 terms of the sequence \(7, 7.7, 7.77, 7.777, \ldots\), we can follow these steps:

### Step 1: Identify the Pattern
The sequence can be expressed as:
- First term: \(7\)
- Second term: \(7.7 = 7 + 0.7\)
- Third term: \(7.77 = 7 + 0.77\)
- Fourth term: \(7.777 = 7 + 0.777\)

We can see that each term can be written as \(7 + \text{(a decimal part)}\).

### Step 2: Factor Out the Common Term
We can factor out the common term \(7\):
\[ S_n = 7 + 7.7 + 7.77 + 7.777 + \ldots \text{ (50 terms)} \]
This can be rewritten as:
\[ S_n = 7 \times (1 + 1.1 + 1.11 + 1.111 + \ldots \text{ (50 terms)}) \]

### Step 3: Simplify the Decimal Part
Now, we need to analyze the series inside the parentheses:
\[ 1 + 1.1 + 1.11 + 1.111 + \ldots \]
This can be expressed as:
\[ 1 + \frac{11}{10} + \frac{111}{100} + \frac{1111}{1000} + \ldots \]

### Step 4: Multiply and Divide by 9
To simplify this, we can multiply and divide by \(9\):
\[ S_n = 7 \times \left( \frac{1 + 1.1 + 1.11 + \ldots}{1} \right) = 7 \times \left( \frac{9}{9} \times (1 + 1.1 + 1.11 + \ldots) \right) \]
This gives us:
\[ S_n = \frac{7}{9} \times (9 + 9.9 + 9.99 + \ldots) \]

### Step 5: Express Each Term
We can express each term as:
- \(9 = 10 - 1\)
- \(9.9 = 10 - 0.1\)
- \(9.99 = 10 - 0.01\)
- \(9.999 = 10 - 0.001\)

So we can rewrite the sum:
\[ S_n = \frac{7}{9} \times \left( (10 - 1) + (10 - 0.1) + (10 - 0.01) + \ldots \text{ (50 terms)} \right) \]

### Step 6: Sum the Terms
Now we can sum the terms:
\[ S_n = \frac{7}{9} \times \left( 10 \times 50 - (1 + 0.1 + 0.01 + \ldots) \right) \]

### Step 7: Calculate the Geometric Series
The series \(1 + 0.1 + 0.01 + \ldots\) is a geometric series with:
- First term \(a = 1\)
- Common ratio \(r = 0.1\)
- Number of terms \(n = 50\)

The sum of a geometric series is given by:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]
Substituting the values:
\[ S_n = \frac{1(1 - (0.1)^{50})}{1 - 0.1} = \frac{1 - (0.1)^{50}}{0.9} \]

### Step 8: Substitute Back
Now substituting back into our equation:
\[ S_n = \frac{7}{9} \times \left( 500 - \frac{1 - (0.1)^{50}}{0.9} \right) \]

### Step 9: Final Calculation
Calculating the final sum:
\[ S_n = \frac{7}{9} \times \left( 500 - \frac{10}{9} + \frac{(0.1)^{50}}{9} \right) \]
This simplifies to:
\[ S_n = \frac{7}{81} \times (4500 - 10 + (0.1)^{49}) \]

### Final Answer
Thus, the sum of the first 50 terms of the sequence is:
\[ S_n = \frac{7}{81} \times (4490 + (0.1)^{49}) \]