A boy charges rupee 1 for first day, rupee 2 for second day and rupee 4 for third and so on. If boy starts work on 1st feb and completed it on 20th feb, then how much amount he get ?

Options

A. `2^20`
B. `2^20-1`
C. `2^19-1`
D. `2^19`

Answer

To find out how much amount the boy earns from 1st February to 20th February, we can follow these steps:

### Step 1: Identify the pattern of earnings
The boy charges:
- Rs. 1 on the 1st day (which is \(2^0\))
- Rs. 2 on the 2nd day (which is \(2^1\))
- Rs. 4 on the 3rd day (which is \(2^2\))
- Rs. 8 on the 4th day (which is \(2^3\))
- And so on...

This means on the \(n\)th day, he charges \(2^{(n-1)}\).

### Step 2: Determine the total number of days worked
The boy works from 1st February to 20th February, which is a total of 20 days.

### Step 3: Set up the formula for the sum of the series
The total amount earned over 20 days can be represented as the sum of a geometric series:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Where:
- \(S_n\) = sum of the first \(n\) terms
- \(a\) = first term of the series
- \(r\) = common ratio
- \(n\) = number of terms

### Step 4: Substitute the values into the formula
Here,
- The first term \(a = 1\) (which is \(2^0\))
- The common ratio \(r = 2\)
- The number of terms \(n = 20\)

Substituting these values into the formula gives:
\[ S_{20} = 1 \cdot \frac{2^{20} - 1}{2 - 1} \]
This simplifies to:
\[ S_{20} = 2^{20} - 1 \]

### Step 5: Calculate \(2^{20}\)
Now, we need to calculate \(2^{20}\):
\[ 2^{20} = 1048576 \]

### Step 6: Final calculation
Now, substituting back into the sum:
\[ S_{20} = 1048576 - 1 = 1048575 \]

### Conclusion
The total amount the boy earns from 1st February to 20th February is **Rs. 1048575**.

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