What is sum to the 100 terms of the series `9+99+999+…?`

Options

A. `(10)/(9)(10^(100)-1)-100`
B. `(10)/(9)(10^(99)-1)-100`
C. `100(100^(10)-1)`
D. `(9)/(100)(10^(100)-1)`

Explanation

To find the sum of the first 100 terms of the series \(9 + 99 + 999 + \ldots\), we can express each term in a more manageable form.

### Step 1: Express each term in a standard form
The series can be rewritten as:
- The first term: \(9 = 10^1 - 1\)
- The second term: \(99 = 10^2 - 1\)
- The third term: \(999 = 10^3 - 1\)

Thus, the \(n\)-th term can be expressed as:
\[ T_n = 10^n - 1 \]

### Step 2: Write the sum of the first 100 terms
Now, we can write the sum \(S_n\) of the first 100 terms:
\[ S_{100} = (10^1 - 1) + (10^2 - 1) + (10^3 - 1) + \ldots + (10^{100} - 1) \]

### Step 3: Simplify the sum
This can be simplified as:
\[ S_{100} = (10^1 + 10^2 + 10^3 + \ldots + 10^{100}) - (1 + 1 + 1 + \ldots + 1) \]
The second part, \(1 + 1 + 1 + \ldots + 1\), has 100 terms, so it equals \(100\).

### Step 4: Calculate the sum of the geometric series
The first part is a geometric series where:
- The first term \(a = 10\)
- The common ratio \(r = 10\)
- The number of terms \(n = 100\)

The sum of a geometric series is given by:
\[ S_n = \frac{a(r^n - 1)}{r - 1} \]
Substituting the values:
\[ S_{100} = \frac{10(10^{100} - 1)}{10 - 1} = \frac{10(10^{100} - 1)}{9} \]

### Step 5: Combine the results
Now, substituting back into our equation for \(S_{100}\):
\[ S_{100} = \frac{10(10^{100} - 1)}{9} - 100 \]

### Step 6: Final expression
Thus, the final expression for the sum of the first 100 terms is:
\[ S_{100} = \frac{10(10^{100} - 1)}{9} - 100 \]