Lesson: Sums of Sequences

Comment on Sums of Sequences

Hi Brent!

Can you please answer this question

http://greprepclub.com/forum/the-sum-of-the-odd-even-integers-1867.html



https://greprepclub.com/forum/the-sum-of-the-odd-even-integers-1867.html

Hi Brent - besides the way that you solved it on the forum, can u provide an alternative way to solve it? Mainly because I'm having a hard time understanding the other solutions that were posted.
greenlight-admin's picture

You bet!

I've added a second/different solution here: https://greprepclub.com/forum/the-sum-of-the-odd-even-integers-1867.html...

Cheers,
Brent

https://greprepclub.com/forum/the-sum-of-the-odd-even-integers-1867.html#p19621

I solved this question by using the formula Sum = n/2(first + last)

For A: n = (199-1+1) = 199/2 = 99.5 ~ 100
Sum = 100/2(1+199) = 10,000

For B: n = (198-2+1) = 197/2 = 98.5 ~ 99
Sum = 99/2(2+198) = 9,900

Is this approach correct?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/the-sum-of-the-odd-even-integers-1867.html

That's a perfectly valid solution. Nice work!

Cheers,
Brent

what lesson does this equation fall under? Because I am still having a hard time understanding how you solved it?
greenlight-admin's picture

The concept required to answer this question falls under the heading Sums of Sequences (video above), but my solution somewhat circumvents the requirement to find any actual sums.

Also note that this is a very tricky question!

Cheers,
Brent

I noticed that you have not mentioned geometric sequences in this video. They aren't that important for gre or they out of scope?
greenlight-admin's picture

Good question.

Geometric sequences are out of scope. That is, you aren't expected to know what a geometric sequence is, and you aren't expected to know how to find individual terms (or the sum) of a geometric sequence.

That said, it's conceivable that a GRE question might describe/define a sequence that is geometric. However, the test-makers would likely construct the question so that previous knowledge of geometric sequences would not give the test-taker an advantage.

Cheers,
Brent

https://greprepclub.com/forum/2-5112.html

In the solutions for the question posted above, looking at the explanations I understand everything until step 2 (or when you factor out a (-2) and leave the positive difference of squares)). Can you explain to me why you factored out a -2, and the steps after?

Thank you!
greenlight-admin's picture

Sorry. It looks like I made an error after factoring out the -2.

I've edited my response (at https://greprepclub.com/forum/2-5112.html#p9241), but I'll explain my rationale nonetheless.

Let's start here: answer = (-2)(2 + 4) + (-2)(6 + 8) + (-2)(10 + 12) + (-2)(14 + 16) + (-2)(18 + 20)

Rather than multiply -2 by each of the sums and then find that entire sum, I see that I can save some time by first factoring out the -2 to get the following:
answer = (-2)[(2 + 4) + (6 + 8) + (10 + 12) + (14 + 16) + (18 + 20)]

From here, I just need to add all of the values inside the square brackets and then multiply the entire sum by -2

Does that help?

Cheers,
Brent

https://greprepclub.com/forum/if-9189.html
can you please elaborate? I didnt understand how you got 21+22?
greenlight-admin's picture

Question/solution link: https://greprepclub.com/forum/if-9189.html#p19449

The given pattern, (-20) + (-19) + (-18) + (-17) + . . . , tells us that the terms are CONSECUTIVE integers.

In my solution, I recognized that, if we keep the sequence going to 20, we get: (-20) + (-19) + (-18) + (-17) + . . . + 19 + 20, which equals ZERO.

So, from that point, if we keep adding more CONSECUTIVE integers, we add 21, then we add 22, etc

We know that (-20) + (-19) + (-18) + (-17) + . . . + 19 + 20 = 0,
So, (-20) + (-19) + (-18) + (-17) + . . . + 19 + 20 + 21 = 0 + 21 = 21
And (-20) + (-19) + (-18) + (-17) + . . . + 19 + 20 + 21 + 22 = 0 + 21 + 22 = 43
And (-20) + (-19) + (-18) + (-17) + . . . + 19 + 20 + 21 + 22 + 23 = 66
And (-20) + (-19) + (-18) + (-17) + . . . + 19 + 20 + 21 + 22 + 23 + 24 = 90

Does that help?

Cheers,
Brent

https://greprepclub.com/forum/the-sum-of-the-odd-even-integers-1867.html

I was wondering if this is a valid approach:

A = 199 numbers/2 = 99.5 = around 100 numbers are even
B = 197 numbers/2 = 98.5 = around 99 numbers are odd

Then use --> sum = n(n+1)/2

So:
A = 100(100+1)/2 = 5050
B = 99(99+1)/2 = 4950

So, A> B
greenlight-admin's picture

Question link: https://greprepclub.com/forum/the-sum-of-the-odd-even-integers-1867.html

This approach can get you into trouble at times.
For example, why did you use 197 for Quantity B?

Also, the sum = n(n+1)/2 applies only to consecutive integers.
It doesn't apply to consecutive EVEN integers or consecutive ODD integers.

See what answer you get if you consider the integers from 1 to 198.
You should find that A < B.
Give it a try.

Cheers,
Brent

Hello !

https://greprepclub.com/forum/the-sum-of-the-odd-even-integers-1867.html
Please, can you explain a little more why you subtract both of them. And why you get positives ones
When I subtract it . I get many minus ones.

Also if it possible can you explain the first approach that is the beginning, in the same link.
Thank you
greenlight-admin's picture

Question link: https://greprepclub.com/forum/the-sum-of-the-odd-even-integers-1867.html...

In my 2nd step, I subtracted all of Quantity B from both quantities so that I could have 0 in Quantity B (makes it very easy to compare)

Notice that, in the sum 1 + 3 + 5 + ... + 197 + 199, we are adding 100 numbers
Conversely, in the sum 2 + 4 + 6 + ... + 196 + 198, we are adding 99 numbers

So, if we subtract (as I believe you did), we get:
(1-2) + (3-4) + (5-6) + . . . .+ (195-196) + (197-198) + 199
= (-1) + (-1) + (-1) + . . . + (-1) + (-1) + 199
= -99 + 199
= 100

Does that help?

Cheers,
Brent

https://greprepclub.com/forum/for-each-integer-n-1-let-a-n-denote-the-sum-2131.html
In this question why is it not solved by using the formula directly [n(n+1)]/2.
The answer is same
greenlight-admin's picture

https://greprepclub.com/forum/for-each-integer-n-1-let-a-n-denote-the-su...

Good question!
My solution assumed that students are unaware of that formula. I think it's useful to see how we can still answer the question without any knowledge of the formula.

That said, I just added a second solution by applying the formula: https://greprepclub.com/forum/for-each-integer-n-1-let-a-n-denote-the-su...

Cheers,
Brent

https://greprepclub.com/forum/if-9189.html
I don't get why you would equal 24 to -2x-4. Can you explain?
Thanks
greenlight-admin's picture

My solution: https://greprepclub.com/forum/if-9189.html#p19449

Let's start here: 21 + 22 + ? + ? + ... + (-2x - 4) = 90
Starting from 21, we want to keep adding consecutive integers until we get a sum of 90.
NOTE: (-2x - 4) is the last consecutive integer in the sum

After testing a few sums, we find that 21 + 22 + 23 + 24 = 90
So the last consecutive integer in the sum is 24
This means -2x - 4 = 24

Does that help?

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