There are fifty successive percentage discounts given in a series of 2%, 4%, 6%, 8%...and so on. What is the net discount?

Options

A. 0.98
B. 25.5
C. 1
D. Infinite

Answer

To find the net discount given a series of successive percentage discounts of 2%, 4%, 6%, 8%, and so on, we can follow these steps:

### Step 1: Understanding the Discounts
The discounts are given in a series: 2%, 4%, 6%, 8%, ..., up to the 50th term. The nth term of this series can be represented as \(2n\), where \(n\) is the term number.

### Step 2: Calculate the Total Discount Percentage
To find the total discount percentage, we need to sum the series of discounts:
\[ \text{Total Discount} = 2 + 4 + 6 + 8 + ... + 100 \]
This is an arithmetic series where:
- First term \(a = 2\)
- Last term \(l = 100\)
- Number of terms \(n = 50\)

The sum \(S_n\) of the first \(n\) terms of an arithmetic series can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (a + l) \]
Substituting the values:
\[ S_{50} = \frac{50}{2} \times (2 + 100) \]
\[ S_{50} = 25 \times 102 = 2550 \]

### Step 3: Calculate the Average Discount
To find the average discount percentage, we divide the total discount by the number of terms:
\[ \text{Average Discount} = \frac{S_{50}}{n} = \frac{2550}{50} = 51 \]

### Step 4: Determine the Net Discount
The net discount is not simply the average discount. Instead, we need to understand that successive discounts compound. The formula for the net price after successive discounts can be represented as:
\[ P' = P \times (1 - d_1) \times (1 - d_2) \times ... \times (1 - d_n) \]
Where \(d_i\) is the decimal form of the discount percentage.

For our case, we can express the net price after all discounts as:
\[ P' = 100 \times (1 - 0.02) \times (1 - 0.04) \times (1 - 0.06) \times ... \times (1 - 0.100) \]

### Step 5: Calculate the Net Price
Calculating this directly can be complex, but we can conclude that as the number of successive discounts increases, the net price approaches zero, indicating that the net discount approaches 100%.

### Conclusion
Thus, the net discount after applying all 50 successive discounts is effectively 100%.

### Final Answer
The net discount is **100%**.

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