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The sum of all odd integers from 1 to 2001 is,

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1002002

1002001

1020120

1122303

answer is B.

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Detailed Solution

We need to find the sum of all odd integers from 1 to 2001.
The odd integers between 1 and 2001 are 1, 3, 5, 7, …….2001 which forms an A.P. because the common difference is same.
We know,

S n = n 2 2 a + n − 1 d .

where,
The number of terms is n.
The first term is a.
The common difference is d.
According to given sequence,

a_{1} = 1 (first term), a_{2} = 3

(2nd term).
Now,
Substitute 1 for

a 1

and 3 for

a 2

d = a 2 − a 1

obtain the common difference d.

d = a 2 − a 1 ⇒ d = 3 − 1 ⇒ d = 2

Now,

a n = a + n − 1 d ⇒ 2001 = 1 + n − 1 2 ⇒ 2000 = n − 1 2

⇒ 2 n − 2 = 2000 ⇒ 2 n = 2002 ⇒ n = 1001

Therefore, the sum will be,

S 1001 = 10012 21 + 1001 − 1 2 ⇒ S 1001 = 10012 2 + 2000 ⇒ S 1001 = 100122002

⇒ S 1001 = 10011001 ⇒ S 1001 = 1002001

The sum of all odd numbers lying between 1 and 2001 is 1002001.
Hence, option 2 is correct.

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