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Medium level - sum of integers

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Mar 05, 2009 1:28 pm

If the sum of the integers from 1 to 100 is 5050, then the sum of the odd integers from 1 to 99 is
(A) 2500
(B) 2525
(C) 2550
(D) 2575
(E) 2600

Brent Hanneson - Creator of GMATPrepNow.com

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Source: Beat The GMAT — Problem Solving | Thu Mar 05, 2009 1:28 pm

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Mar 05, 2009 4:42 pm

It's dinner time, so I better post my solution before I go.
The answer is A.

Brent Hanneson - Creator of GMATPrepNow.com

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pakaskwa: Master | Next Rank: 500 Posts; Posts: 107; Joined: Wed Mar 04, 2009 4:39 am; Location: Vancouver; Thanked: 9 times; GMAT Score:750

by pakaskwa » Thu Mar 05, 2009 7:25 pm

For any arithmetic progression, the sum is

S = n(a1+an)/2

Where n is the number of sequence. For the question, there are 50 numbers from 1,3,5,... to 99, half of 100.
a1=1, a50=99,

So S=50(1+99)/2=2500

But the interesting part is that, why the sum is not half of 5050?

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satish.nagdev: Master | Next Rank: 500 Posts; Posts: 130; Joined: Sat Dec 29, 2007 5:35 am; Location: SGP; Thanked: 5 times

by satish.nagdev » Fri Mar 06, 2009 12:38 am

pakaskwa wrote: But the interesting part is that, why the sum is not half of 5050?

because every even integer falls after odd and its value is +1 of the odd number, so 50 even numbers in set [1 to 100] result in sum of even numbers = sum of odd numbers +50

its like
1 2
3 4
....
.....
99 100

and thanks pakaskwa your formula for finding sum was quick one

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pakaskwa: Master | Next Rank: 500 Posts; Posts: 107; Joined: Wed Mar 04, 2009 4:39 am; Location: Vancouver; Thanked: 9 times; GMAT Score:750

by pakaskwa » Fri Mar 06, 2009 3:08 pm

That makes sense! I didn't think enough.

By the way, I think Brent's idea is to let people know that, if they don't remember the formula, how they can get the correct numbers.

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logitech: Legendary Member; Posts: 2134; Joined: Mon Oct 20, 2008 11:26 pmFollowed by: 25 members; Thanked: 237 times; GMAT Score:730

Re: Medium level - sum of integers

by logitech » Sun Mar 08, 2009 3:39 pm

Brent Hanneson wrote:If the sum of the integers from 1 to 100 is 5050, then the sum of the odd integers from 1 to 99 is
(A) 2500
(B) 2525
(C) 2550
(D) 2575
(E) 2600

1+2+...+n = (n)x(n+1)/2

1+2+3+4+5+6+7+8+9+10 = (10)(10+1)/2 = 55

1+3+5+7+9 = 25

2+4+6+8+10 = 2( 1+2+3+4+5) = 2x(5)x(5+1)/2 = 30

I used this simple set to explain you the logic behind the calculation.

So:

2+4+6+..100 = 2( 1+2+3...+50) = 2x(50)(51)/2 = 50x51

So the sum of odd numbers is:

5050 - ( SUM OF EVEN ) = (101)x(50) - (50) x (51 ) = 50x50 = 2500

LGTCH
---------------------
"DON'T LET ANYONE STEAL YOUR DREAM!"

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deepoe: Master | Next Rank: 500 Posts; Posts: 151; Joined: Wed Feb 11, 2009 2:45 am; Thanked: 3 times

by deepoe » Thu Apr 16, 2009 2:59 am

I thought this

1+3+5+7+9 = 25

First digit = 10 numbers
Second digit = 10 numbers

So 10x10x25 = 2500

But is this a good way to calculate it?

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