Statement -1 For all natural numbers n , `0.5+0.55+0.555+......` upto n terms `=(5)/(9){n-(1)/(9)(1-(1)/(10^n))}` ,
Statement-2 `a+ar+ar^2+....+ar^(n-1)=(a(1-r^n))/((1-r))`, for `0lt r lt 1` .

Options

A. (a)Statement -1 is true , Statement -2 is true, Statement -2 is correct explanation for Statement -1
B. (b)Statement -1 is true , Statement -2 is true , Statement -2 is not correct explanation for Staement -1
C. (c)Statement -1 is true , Statement -2 is false
D. (d)Statement -1 is false , Statement - 2 is true.

Explanation

To solve the problem, we will use mathematical induction to prove Statement 1, which states that for all natural numbers \( n \):

\[ 0.5 + 0.55 + 0.555 + \ldots \text{ (up to n terms)} = \frac{5}{9} \left( n - \frac{1}{9} \left( 1 - \frac{1}{10^n} \right) \right) \]

### Step 1: Base Case
First, we check the base case when \( n = 1 \):
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