What is the difference between the sum of the first 2003 positive even

Question

What is the difference between the sum of the first 2003 positive even integers and the sum of the first 2003 positive odd integers ?

(A) 0
(B) 1
(C) 2
(D) 2003
(E) 4006

Archit3110 · Answer

n even integers : 2+4+6+...+4006
and odd intgers : 1+3+5+...+4005
difference would be of 2003 nos only
IMO D

jfranciscocuencag · Answer

In this case we can just solve it by theory, but it is important to learn the following formulas:

Sum of the first n even integers:   n(n+1)  
Sum of the first n odd integers:   n^2

So...

((2003(2003+1))-(2003^2))

((2003*2004)-2003^2)

2003((1)(2004)-2003)

2003(2004-2003)

2003(1)

2003

D

ScottTargetTestPrep · Answer

If we pair up each odd and even integer, we have:

1,2

3,4

5,6

…..

4,005, 4,006

We see that each even number is 1 more than its odd counterpart, and since there are 2003 such pairs, the difference between the sum of the even integers and the sum of the odd odd integers is 2003.

Answer: D

HarshaBujji · Answer

Solved in 0.08

To tackle such problem always choose small subsets of same question.

I mean what's the difference in first 1 odd and first 1 even.

2-1=1

Iteeation 2: 
First 2 even - first 2 odd
2-1+4-3 = 2

First 3 even - first 3 odd
2-1+4-3+6-5= 3

Here you can observe the pattern right.

In maths patterns are always formed.

Hence for 2003 , ans 2003.

Posted from my mobile device

MathRevolution · Answer

Let's have odd numbers of some even positive integers and some odd positive integers:

First 3 even positive integers : 2, 4, 6 
Sum = 2 + 4 + 6 = 12

First 3 odd positive integers : 1, 3, 5 
Sum = 1 + 3 + 5 = 9

Difference between first 3 even and odd positive integers: 12 - 9 = 3

Similarly, If we take the first 5 even integers: 2, 4, 6, 8, 10 
Sum = 2 + 4 + 6 + 8 + 10 = 30

First 5 odd positive integers : 1, 3, 5, 7, 9 
Sum = 1 + 3 + 5 + 7 + 9 = 25

Difference between first 5 even and odd positive integers: 30 - 25 = 5

Hence, the difference between the first 2003 even and odd positive integers: 2003

Answer D

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What is the difference between the sum of the first 2003 positive even

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What is the difference between the sum of the first 2003 positive even

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