Find the sum upto n terms of the series `a+aa+aaa+aaaa+"......",AA a in N " and "1 le a le 9.`

Explanation

To find the sum up to n terms of the series \( S = a + aa + aaa + aaaa + \ldots \) where \( a \) is a natural number and \( 1 \leq a \leq 9 \), we can express each term in the series in a more manageable form.

### Step-by-Step Solution:

1. **Understanding the Terms**:
The first term is \( a \), the second term is \( aa = 10a + a = 11a \), the third term is \( aaa = 100a + 10a + a = 111a \), and so on. Each term can be represented as:
\[
T_n = \underbrace{111\ldots1}_{n \text{ ones}} \cdot a
...