Problem Solving

Hi,

this question is from my gmat prep test, I actually got it right by chance, but would be nice to have a logical explanation.

The sum of the first 50 even positive integers is 2550. What is the sum of the even integers from 102 to 200 inclusive?

A. 5100
B.7550
C. 10100
D. 15500
E. 20100

Thanks in advance!

Is it B?

yes, the spoiler function didn't work, that's why I differentiated it from the other answer choices

How did you get there?

The no. of terms (n) = 102+200/2 +1 = 50

Then using the sum of arithmetic sequence formula:

S = n/2 (2a + (n-1)d), where a is the 1st term, d is the difference between the terms, and n is the no. of terms

S= 50/2 (2*102 + (49)2)

S= 25 (204 + 98)

S = 7550

Great question and discussion, guys! A few kind of neat things here:

1) GMAT questions are very rarely inefficient...you can almost always use all the given information in some way to help you find an elegant solution. Here, they give you that "the sum of the first 50 is 2550" for a reason. Those numbers are:

2, 4, 6....98, 100

Well, the even integers 102 to 200 should look pretty similar:

102, 104, 106...198, 200

Each number in the second set corresponds directly to a number in the first set, plus 100. So you're essentially taking the original sum (2550) and adding 100 to it 50 times (once for each number). 2550 + 100*50 = 2550 + 5000 = 7550.


2) Or, doing it without using the first sentence of the question, MBA.Aspirant is right. But if you don't think you can (or you just don't want to) memorize that formula, think of it this way - if we know the average and we know the number of terms, then we can employ this easy-to-remember formula:

Average = Sum of terms / Number of terms

and apply it to figure out the sum of the terms:

Average * Number of terms = Sum of terms (which is what we're looking for)

It's an evenly spaced set, so the average will be the middle number: (102 + 200) / 2 = 302/2 = 151.

And we know that there are 50 terms, so we'll multiply 151*50 to get 7550.

Note that this method uses essentially the same methodology that MBA.Aspirant's formula does, but I bring it up because it doesn't require you to memorize a brand-new formula. You should already know the average formula, and this just uses it in another way to solve a totally different type of problem. The GMAT is all about leveraging your assets, so if you can find a way to use something you already know to solve a brand new problem, you're doing exactly what they want to see from you.

Thanks so much!!! The second approach is actually very helpful...can't wait to see questions like this from now on