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What is the sum of all the odd positive integers less than $100$ ?

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2600

2500

2400

2300

answer is B.

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Detailed Solution

The given possible odd positive integers

1, 3, 5, 7, 9, 11 . . . . . . . . . . 99

.
The numbers form an AP with first term 1 and common difference 2. So, we get
The first term,

= 1

.
Common difference d = 2.
The total no. of terms

= 50 .

The sum of first n terms of an AP is given as,

S_{n} = \frac{n}{2} [2 a + (n - 1) d]

On substituting the values, we get

S_{50} = \frac{50}{2} [2 \times 1 + (50 - 1) 2]

\Rightarrow S_{50} = 25 (2 + 98)

\Rightarrow S_{50} = 50 + 2450

\Rightarrow S_{50} = 2500

Therefore, the sum of all odd positive numbers less than 100 is 2500.
Hence, the correct option is 2.

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