The sum of the first 50 positive even integers is 2,550.

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Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pm; Thanked: 6 times; Followed by:5 members

The sum of the first 50 positive even integers is 2,550.

by Vincen » Fri Sep 29, 2017 6:14 pm

The sum of the first 50 positive even integers is 2,550. What is the sum of the first 50 odd integers?

(A) 2,000
(B) 2,497
(C) 2,500
(D) 2,501
(E) 2,549

The OA is C.

What is the relation between the sum of the even and the sum of the odd integers? I already know that the answer should be an even number. So options (B), (D) and (E) are out.

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Source: Beat The GMAT — Problem Solving | Fri Sep 29, 2017 6:14 pm

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by Brent@GMATPrepNow » Fri Sep 29, 2017 7:13 pm

Vincen wrote:The sum of the first 50 positive even integers is 2,550. What is the sum of the first 50 odd integers?

(A) 2,000
(B) 2,497
(C) 2,500
(D) 2,501
(E) 2,549

The sum of the first 50 positive EVEN integers is 2,550
So, 2 + 4 + 6 + . . . + 98 + 100 = 2550

What is the sum of the first 50 odd integers?
So, we want the value of 1 + 3 + 5 + . . . + 97 + 99

IMPORTANT: Notice that:
1 = 2 - 1
3 = 4 - 1
5 = 6 - 1
.
.
.
97 = 98 - 1
99 = 100 - 1

So, 1 + 3 + 5 + . . . + 97 + 99 = (2 - 1) + (4 - 1) + (6 - 1) + . . . + (98 - 1) + (100 - 1)
= [2 + 4 + 6 + . . . + 98 + 100 ] - 1 - 1 - 1 - 1 . . . . -1 - 1
= [2550 ] - 1 - 1 - 1 - 1 . . . . -1 - 1

ASIDE: How many 1s are we subtracting from 2550 ?
Well, there are 50 even integers from 2 to 100, so we are subtracting fifty 1's
So, we get:
1 + 3 + 5 + . . . + 97 + 99 = [2550 ] - 1 - 1 - 1 - 1 . . . . -1 - 1
= [2550 ] - 50
= 2500

Answer: C

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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

by Brent@GMATPrepNow » Fri Sep 29, 2017 8:40 pm

Vincen wrote:The sum of the first 50 positive even integers is 2,550. What is the sum of the first 50 odd integers?

(A) 2,000
(B) 2,497
(C) 2,500
(D) 2,501
(E) 2,549

Note that we can also solve this question WITHOUT using the fact that the sum of the first 50 positive even integers is 2,550

We need only look for a pattern in the sum of ODD integers.
1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
1 + 3 + 5 + 7 + 9 + 11 = 36

Notice that each sum is the square of an integer (e.g., 4 = 2Â², 9 = 3Â², 16 = 4Â², etc)
So, it looks like the sum of the first 50 integers will be the square of an integer.
Since 2500 is the only square of an integer, it is most likely the correct answer.

Alternatively, we might recognize that:
The sum of the first 1 ODD integer = 1Â²
The sum of the first 2 ODD integers = 2Â²
The sum of the first 3 ODD integers = 3Â²
The sum of the first 4 ODD integers = 4Â²
.
.
.
So, the sum of the first 50 ODD integers = 50Â² = 2500

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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jbryant62: Junior | Next Rank: 30 Posts; Posts: 22; Joined: Wed Nov 23, 2016 4:04 pm; Thanked: 1 times

by jbryant62 » Thu Oct 05, 2017 11:33 am

Vincen wrote:The sum of the first 50 positive even integers is 2,550. What is the sum of the first 50 odd integers?

(A) 2,000
(B) 2,497
(C) 2,500
(D) 2,501
(E) 2,549

The OA is C.

What is the relation between the sum of the even and the sum of the odd integers? I already know that the answer should be an even number. So options (B), (D) and (E) are out.

the sum of any set of evenly spaced set of numbers is equal to the 1st number plus the last number divided by 2 then multiplied by the number of numbers in the set. so in the example above. 2+100/2=51 there are 50 numbers in the set. 50*51=2550. for odd its 1+99/2=50. again 50 numbers in the set. 50*50=2500

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EconomistGMATTutor: GMAT Instructor; Posts: 555; Joined: Wed Oct 04, 2017 4:18 pm; Thanked: 180 times; Followed by:12 members

by EconomistGMATTutor » Fri Oct 06, 2017 10:24 am

Vincen wrote:The sum of the first 50 positive even integers is 2,550. What is the sum of the first 50 odd integers?

(A) 2,000
(B) 2,497
(C) 2,500
(D) 2,501
(E) 2,549

The OA is C.

What is the relation between the sum of the even and the sum of the odd integers? I already know that the answer should be an even number. So options (B), (D) and (E) are out.

Hi Vincen,
Let's take a look at your question.
Given is that the sum of first 50 positive integers os 2,550.
2 + 4 + 6 + 8 + ... + 100 = 2,550 --- (i)
We are asked to find the sum of first 50 odd integers.
If we subtract 1 from each of the even number in (i), we will get the sum of first 50 odd numbers. Like
(2-1) + (4-1) + (6-1) + (8-1) + ... + (100-1)

Seaparate all the 50 1s, we get
=2 + 4 + 6 + 8 + ... + 100 - 50 (50 1's add upto 50)
=2,550 - 50 (Since 2 + 4 + 6 + 8 + ... + 100 = 2,550 from (i))
= 2,500

Therefore, Option C is correct.

I am available if you'd like any followup.

GMAT Prep From The Economist
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Scott@TargetTestPrep: GMAT Instructor; Posts: 8086; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

Re: The sum of the first 50 positive even integers is 2,550.

by Scott@TargetTestPrep » Fri Jan 24, 2020 3:16 pm

Vincen wrote: ↑
Fri Sep 29, 2017 6:14 pm
The sum of the first 50 positive even integers is 2,550. What is the sum of the first 50 odd integers?

(A) 2,000
(B) 2,497
(C) 2,500
(D) 2,501
(E) 2,549

The OA is C.

What is the relation between the sum of the even and the sum of the odd integers? I already know that the answer should be an even number. So options (B), (D) and (E) are out.

Solution:

We can use the formula sum = average x quantity, and if we have an evenly spaced set (notice that a set of consecutive odd integers is an evenly spaced set), average = (first number + last number)/2.

Therefore, since the first odd positive integer = 1 and the last (or in this case, the 50th) odd positive integer = 99, we have average = (1 + 99)/2 = 50. Since quantity = 50, then sum = average x quantity = 50 x 50 = 2500.

Alternate Solution:

Let’s say we don’t know any formulas involving the sum of an evenly spaced set. Let’s just use the given in the problem: “The sum of the first 50 positive even integers is 2,550.” In other words, we are given the sum of 2, 4, 6, …, 100 and we are asked to find the sum of the first 50 positive odd integers, which is the sum of 1, 3, 5, …, 99. Now notice that 1 is 1 less than 2, 3 is 1 less than 4, 5 is 1 less than 6, …, and 99 is 1 less than 100. In other words, each number in the set of the first 50 positive odd integers is 1 less than its counterpart in the set of the first 50 positive even integers. Since there are 50 numbers in each set, then the sum of the first 50 positive odd integers should be 50 less than the sum of the first 50 positive even integers. That is, the sum of the first 50 positive odd integers = 2550 - 50 = 2500.

Answer: C

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