Lesson: Sums of Sequences

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Can you please assist with this question in the 2016 OG book: Question #189 - 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? I don't understand the explanation in the back of the book. The correct answer is B. 20,200, but I'm unsure of how this answer was reached. I got 20,301. I got the sum of the even integers (100-300) which is 201. Then I used the formula to get 201*(201+1)/2 and got 20,301. I don't understand how the answer is 20,200. Thanks!
gmat-admin's picture

Hi Yvonne,

Let's start with "I got the sum of the even integers (100-300) which is 201." If we ADD a bunch of even integers, the sum will be EVEN (201 is odd). Also, the SUM will be a lot bigger than 201 :-)

I believe you are saying that there are 201 integers from 100 to 300 inclusive. That part is correct. However, there are two problems with this information:

1) The question is asking for the EVEN integers from 100 to 300 inclusive. Your calculation (of 201) just tells us the integers (both odd AND even).

2) The formula n(n+1)/2 tells us the sum of the first n positive integers. For example, to find the sum 1+2+3+4+5+6, we see that n = 6 in this case. So, using the formula, the sum = (6)(6+1)/2 = 21. You are taking the number 201 (which represents something totally different), and you are trying to use the formula that finds the sum of the first n integers.

I provide 3 different solutions to that question here: http://www.beatthegmat.com/consecutive-integers-t275599.html


Thanks Brent!

I see what I did wrong. I read all three of your solutions and it definitely makes more sense now.

Do you offer solutions to all of the OG 2016 questions on the beatthegmat.com website? Or just a select number of questions? I find the OG explanations to the problems make absolutely no sense at times.
gmat-admin's picture

I have answered quite a few of the OG2016 questions on both Beat The GMAT and GMAT Club, but not all of them.

term(n) = 1/x - 1/(x+1)

What is the sum of the first 100 terms of this sequence?

A) 0
B) 1/101
C) 99/100
D) 100/101
E) 1
Sir can we solve the problem this problem by n(n+1)/2 formula
gmat-admin's picture

No, that formula won't help us here. Instead, we must recognize that almost all of our fractions cancel out. Here's what I mean....

1st term = 1/1 - 1/(1 + 1) = 1/1 - 1/2
2nd term = 1/2 - 1/(2 + 1) = 1/2 - 1/3
3rd term = 1/3 - 1/(3 + 1) = 1/3 - 1/4
4th term = 1/4 - 1/(4 + 1) = 1/4 - 1/5
99th term = 1/99 - 1/(99 + 1) = 1/99 - 1/100
100th term = 1/100 - 1/(100 + 1) = 1/100 - 1/101

So, the SUM = (1/1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) + . . . + (1/99 - 1/100) + (1/100 - 1/101)

We can that the 2nd term in each bracket cancels out with the 1st term in the next bracket.

So, the SUM = 1/1 - 1/(100 + 1) = 1 - 1/101 = 100/101

Answer: D

Brent, your story about Carl Gauss truly amazing! Now I do not need to memorize formula! Thanks for your lesson here.

Anyway, I tried using this approach to answer several question. It seems that Gauss' approach can be used to solve another problem such as calculate the SUM of consecutive multiple 5 starts from 4 and ends in 69. The tricky one maybe in calculate the number of integer there. I found that total number is 14, and the sum is 511.

It is true that we can use it to solve other variation of this problems?

Once again, thank you!
gmat-admin's picture

You are correct; we can apply Gauss' concept to a variety of questions.

Hi Brent,

we don't consider zero as positive number but in below question we consider it.

What is the sum of first 10 non-negative even integers?


If question had asked us to calculate the sum of the first 10 even numbers, will we still consider zero?
gmat-admin's picture

Question link: https://gmatclub.com/forum/what-is-the-sum-of-first-10-non-negative-even...

Be careful; this is a very common trap on the GMAT.

If I say that k is a NON-NEGATIVE integer. This does NOT mean that k must be a positive integer (many students will make this conclusion).

A non-negative number is a number that is not negative. Since 0 is not negative, we can say that 0 is a non-negative number.

ASIDE: 0 is neither positive nor negative.
Your 2nd question: "If the question had asked for the sum of first 10 even numbers, will we still consider zero?"

An official GMAT question would never ask us to determine the sum of the first 10 even numbers, because even numbers are without beginning and end.

EVEN integers: . . . -8, -6, -4, -2, 0, 2, 4, 6, 8, . . .
The ". . . " means the pattern continues forever.

That said, if the question asked "What is the sum of first 10 POSITIVE even numbers?", then we'd find the following sum: 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20

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