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Sum of even Numbers OG 12: 157

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by aplavakarthik » Sun Aug 07, 2011 11:57 pm
For any +ve integer n, the sum of the first n +ve integers equals n(n+1)/2. What is the sum of all the even integers between 99 and 300.
a.10,100
b.20,200
c.22,650
d.40,200
e.45150

b

by subtracting the sum of first 99 and sum of first 300. it is very clear that we can eliminate a,d,e.
I had to choose between b and c.
the option c is to from half of the value obtained above. so i choose b.

Is there is any better way of doing this?

Is there is a way to find sum of even numbers?
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by Anurag@Gurome » Mon Aug 08, 2011 12:16 am
aplavakarthik wrote:For any +ve integer n, the sum of the first n +ve integers equals n(n+1)/2. What is the sum of all the even integers between 99 and 300
There is a much better way to solve this without using the given formula.

For any set of uniformly spaced integers, the average of the integers = (Largest integer + Smallest integer)/2

In this case, all the even integers between 99 and 300, i.e. from 100 to 300 are uniformly spaced. Hence, average of them = (100 + 300)/2 = 200

Now, sum of all these integers = (Average)*(Number of integers in the range) = 200*[(300 - 100)/2 + 1] = 200*101 = 20200

The correct answer is B.
Last edited by Anurag@Gurome on Mon Aug 08, 2011 12:40 am, edited 1 time in total.
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by Touseef » Mon Aug 08, 2011 12:39 am
Thats an Excellent Approach.

There is a small typo error:

sum of all these integers = (Average)*(Number of integers in the range) = 200*[(300 - 100)/2 + 1] = 200*101 = 20200
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by GmatKiss » Mon Aug 08, 2011 12:42 am
Hi Anurag,

In this case, all the even integers between 99 and 300, i.e. from 100 to 300 are uniformly spaced.

How is numbers between 99 and 300 is 100,101.....299,300 ?
How do we omit 99 and include 300 in this case?
Please help!
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by Touseef » Mon Aug 08, 2011 12:46 am
99 is not even.Hence,omitted
300 is even.Hence,included
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by Anurag@Gurome » Mon Aug 08, 2011 12:47 am
GmatKiss wrote:How is numbers between 99 and 300 is 100,101.....299,300 ?
Can you elaborate your query with a bit more words?
GmatKiss wrote:How do we omit 99 and include 300 in this case?
We omit 99 as it is not an even integer and hence do not contribute to the sum. And we include 300 as it is even.
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by GMATGuruNY » Mon Aug 08, 2011 2:46 am
As Anurag noted, the problem above is solved more easily if we ignore the formula given.

A few problems about evenly spaced integers:

https://www.beatthegmat.com/for-any-posi ... 59401.html

https://www.beatthegmat.com/consecutive- ... 73388.html

https://www.beatthegmat.com/sum-of-even- ... 85000.html

https://www.beatthegmat.com/depreciation-t88719.html

https://www.beatthegmat.com/how-many-int ... 88553.html

https://www.beatthegmat.com/tricky-one-t22481.html
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