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Q.

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


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a

1002002

b

1002001

c

1020120

d

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 2a+ n−1 d .  
where,
The number of terms is n.
The first term is a.
The common difference is d.
According to given sequence, a1=1(first term), a2=3(2nd term).
Now,
Substitute 1 for a 1   and 3 for a 2   in 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  
  ⇒2n−2=2000 ⇒2n=2002 ⇒n=1001  
Therefore, the sum will be,
S 1001 = 1001 2 2 1 + 1001−1 2 ⇒ S 1001 = 1001 2 2+2000 ⇒ S 1001 = 1001 2 2002   ⇒ S 1001 =1001 1001 ⇒ 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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