What is the value of `(1)/((0.1)^(2))+(1)/((0.01)^(2))+(1)/((0.2)^(2))+(1)/((0.02)^(2))+(1)/((0.5)^(2))+(1)/((0.05)^(2))`

Options

A. 2929
B. 10504
C. 12625
D. 13029

Answer

To find the value of the expression

\[ \frac{1}{(0.1)^{2}} + \frac{1}{(0.01)^{2}} + \frac{1}{(0.2)^{2}} + \frac{1}{(0.02)^{2}} + \frac{1}{(0.5)^{2}} + \frac{1}{(0.05)^{2}}, \]

we will simplify each term step by step.

### Step 1: Simplify each term
1. Calculate \((0.1)^{2}\):
\[ (0.1)^{2} = 0.01 \implies \frac{1}{(0.1)^{2}} = \frac{1}{0.01} = 100. \]

2. Calculate \((0.01)^{2}\):
\[ (0.01)^{2} = 0.0001 \implies \frac{1}{(0.01)^{2}} = \frac{1}{0.0001} = 10000. \]

3. Calculate \((0.2)^{2}\):
\[ (0.2)^{2} = 0.04 \implies \frac{1}{(0.2)^{2}} = \frac{1}{0.04} = 25. \]

4. Calculate \((0.02)^{2}\):
\[ (0.02)^{2} = 0.0004 \implies \frac{1}{(0.02)^{2}} = \frac{1}{0.0004} = 2500. \]

5. Calculate \((0.5)^{2}\):
\[ (0.5)^{2} = 0.25 \implies \frac{1}{(0.5)^{2}} = \frac{1}{0.25} = 4. \]

6. Calculate \((0.05)^{2}\):
\[ (0.05)^{2} = 0.0025 \implies \frac{1}{(0.05)^{2}} = \frac{1}{0.0025} = 400. \]

### Step 2: Combine the results
Now, we can combine all the simplified terms:
\[ 100 + 10000 + 25 + 2500 + 4 + 400. \]

### Step 3: Perform the addition
1. Add \(100 + 10000 = 10100\).
2. Add \(10100 + 25 = 10125\).
3. Add \(10125 + 2500 = 12625\).
4. Add \(12625 + 4 = 12629\).
5. Finally, add \(12629 + 400 = 13029\).

### Final Result
Thus, the value of the expression is:
\[ \boxed{13029}. \]