The value of `1+1/(4xx3)+1/(4xx3^2)+1/(4xx3^3)`upto four places of decimals is

Options

A. 1.1202
B. 1.1203
C. 1.1204
D. none of these

Answer

To solve the expression \(1 + \frac{1}{4 \times 3} + \frac{1}{4 \times 3^2} + \frac{1}{4 \times 3^3}\) step by step, we can follow these steps:

### Step 1: Write down the expression
We start with the expression:
\[ 1 + \frac{1}{4 \times 3} + \frac{1}{4 \times 3^2} + \frac{1}{4 \times 3^3} \]

### Step 2: Calculate the individual fractions
Calculate each term in the expression:
- The first term is \(1\).
- The second term is \(\frac{1}{4 \times 3} = \frac{1}{12}\).
- The third term is \(\frac{1}{4 \times 3^2} = \frac{1}{4 \times 9} = \frac{1}{36}\).
- The fourth term is \(\frac{1}{4 \times 3^3} = \frac{1}{4 \times 27} = \frac{1}{108}\).

### Step 3: Find a common denominator
The least common multiple (LCM) of the denominators \(12\), \(36\), and \(108\) is \(108\). We will convert each fraction to have this common denominator:
- \(1 = \frac{108}{108}\)
- \(\frac{1}{12} = \frac{9}{108}\) (since \(1 \times 9 = 9\) and \(12 \times 9 = 108\))
- \(\frac{1}{36} = \frac{3}{108}\) (since \(1 \times 3 = 3\) and \(36 \times 3 = 108\))
- \(\frac{1}{108} = \frac{1}{108}\)

### Step 4: Combine the fractions
Now we can add all the fractions together:
\[ \frac{108}{108} + \frac{9}{108} + \frac{3}{108} + \frac{1}{108} = \frac{108 + 9 + 3 + 1}{108} = \frac{121}{108} \]

### Step 5: Convert to decimal
Now, we need to convert \(\frac{121}{108}\) to a decimal:
\[ \frac{121}{108} \approx 1.12037037037 \]
Rounding this to four decimal places gives us:
\[ 1.1204 \]

### Final Answer
Thus, the value of the expression \(1 + \frac{1}{4 \times 3} + \frac{1}{4 \times 3^2} + \frac{1}{4 \times 3^3}\) up to four decimal places is:
\[ \boxed{1.1204} \]