Given that `1+2+3+……….x=(x(x+1))/2` then `1+3+5+…..+99` is equal to

Options

A. 2250
B. 2500
C. 2525
D. 3775

Answer

To solve the problem of finding the sum of the series \(1 + 3 + 5 + \ldots + 99\), we can follow these steps:

### Step 1: Understand the Series
The series \(1 + 3 + 5 + \ldots + 99\) consists of the first 50 odd numbers. We can express this series in terms of a known formula.

### Step 2: Use the Sum of First n Odd Numbers Formula
The sum of the first \(n\) odd numbers is given by the formula:
\[ S_n = n^2 \]
where \(n\) is the number of terms.

### Step 3: Determine the Number of Terms
In our series, the last term is 99. The odd numbers can be represented as:
- The first odd number is 1
- The second odd number is 3
- The third odd number is 5
- ...
- The \(n\)-th odd number is given by the formula \(2n - 1\).

To find \(n\) when the last term is 99:
\[ 2n - 1 = 99 \\ 2n = 100 \\ n = 50 \]
So, there are 50 terms in our series.

### Step 4: Calculate the Sum
Using the formula for the sum of the first \(n\) odd numbers:
\[ S_{50} = 50^2 = 2500 \]

### Final Answer
Thus, the sum \(1 + 3 + 5 + \ldots + 99\) is equal to **2500**.

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