Integers..

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Elena89: Senior | Next Rank: 100 Posts; Posts: 35; Joined: Sun Jul 31, 2011 9:39 pmFollowed by: 1 member; Thanked: 2 times

by Elena89 » Mon Dec 12, 2011 7:00 am

For any positive integer n, the sum of the first n positive integers equals [n(n+1)]/2. What is the sum of all the even integers between 99 and 301 ?

(A) 10,100
(B) 20,200
(C) 22,650
(D) 40,200
(E) 45,150

Source: OG-12

[spoiler]OA: (B) [/spoiler]

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Source: Beat The GMAT — Problem Solving | Mon Dec 12, 2011 7:00 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Dec 12, 2011 7:21 am

I posted a solution here:

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neelgandham: Community Manager; Posts: 1060; Joined: Fri May 13, 2011 6:46 amFollowed by: 52 members; Location: Utrecht, The Netherlands; Thanked: 318 times

by neelgandham » Mon Dec 12, 2011 7:26 am

Hi Elena,

The problem can be solved using different methods. Here are a couple of them.

Method 1)

Consecutive even integers are in Arithmetic Progression, where the common difference of the series is 2, First term 100 and Last term 300.

nth term in an Arithmetic Progression = First term +(n-1)*(Common difference)
300 = 100 + (n-1)*2 => n = 101
Sum of n terms = (n/2)*(first term + last term) = (101/2)*(100+300) = 20200

Method 2)

sum of all the even integers between 99 and 301 = 100+102+104+....300
= 2*(50+51+52+...150)
= 2*((1+2+3+4+...150)-(1+2+3+.....49))
= 2*((0.5*150*151)-(0.5*49*50))
= ((150*151)-(49*50))
= (50*((151*3)-49))
= 50*404
= 20200

Anil Gandham
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Elena89: Senior | Next Rank: 100 Posts; Posts: 35; Joined: Sun Jul 31, 2011 9:39 pmFollowed by: 1 member; Thanked: 2 times

by Elena89 » Mon Dec 12, 2011 7:30 am

Thanks Mitch.. Reading your post made me realize that I was using the right formula with just a tiny mistake!

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ankush123251: Senior | Next Rank: 100 Posts; Posts: 49; Joined: Thu Feb 17, 2011 1:36 amFollowed by: 2 members; Thanked: 8 times

by ankush123251 » Thu Dec 15, 2011 2:04 am

Use the AP formula directly for sum with a1 = 100,an = 300 and d =2.
With this info find n and the Sn = n/2 * (a1 + an).

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