Evaluate : 9 + 99 + 999 + ........... upto n terms.

Answer

To evaluate the expression \( 9 + 99 + 999 + \ldots \) up to \( n \) terms, we can follow these steps:

### Step 1: Rewrite the terms
We can express each term in a more manageable form:
- The first term is \( 9 = 10^1 - 1 \)
- The second term is \( 99 = 10^2 - 1 \)
- The third term is \( 999 = 10^3 - 1 \)
- Continuing this pattern, the \( k \)-th term can be expressed as \( 10^k - 1 \).

Thus, we can rewrite the entire sum as:
\[ 9 + 99 + 999 + \ldots + \text{(up to n terms)} = (10^1 - 1) + (10^2 - 1) + (10^3 - 1) + \ldots + (10^n - 1) \]

### Step 2: Combine the terms
Now, we can combine the terms:
\[ = (10^1 + 10^2 + 10^3 + \ldots + 10^n) - n \]
Here, \( n \) is subtracted because we have \( n \) terms of \(-1\).

### Step 3: Sum the geometric series
The sum \( 10^1 + 10^2 + 10^3 + \ldots + 10^n \) is a geometric series where:
- The first term \( a = 10 \)
- The common ratio \( r = 10 \)
- The number of terms \( n \)

The formula for the sum of the first \( n \) terms of a geometric series is:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Substituting the values:
\[ S_n = 10 \frac{10^n - 1}{10 - 1} = 10 \frac{10^n - 1}{9} \]

### Step 4: Substitute back into the equation
Now we substitute this back into our expression:
\[ = \left(10 \frac{10^n - 1}{9}\right) - n \]

### Step 5: Final expression
Thus, the final expression for the sum \( 9 + 99 + 999 + \ldots \) up to \( n \) terms is:
\[ = \frac{10(10^n - 1)}{9} - n \]