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What is the sum of all $2$ -digit odd numbers?

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2475

2530

4905

5049

answer is A.

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Detailed Solution

The given odd two-digit numbers are

11, 13, 15 . . . 99 .

The numbers from an AP with first term 11 and common difference 2. So, we get
First term,

a = 11

,
Common difference,

d = 2

.
Let there be n terms.
So, we get

a_{n} = 99

.
We know that

n^{th}

term of an AP is given as,

a_{n} = a + (n - 1) d

\Rightarrow a + (n - 1) d = 99

\Rightarrow 11 + (n - 1) 2 = 99

\Rightarrow n = 45

We know that sum of first n terms of an AP is given as,

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

So, the sum of 45 terms will be,

S_{45} = \frac{45}{2} [11 + 99]

\Rightarrow S_{45} = 45 \times 55

\Rightarrow S_{45} = 2475

Therefore, the sum of odd two-digit numbers is 2475.
Hence, the correct option is 1.

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