Want a 51 on Quant? (GMAT Math Blog 1/28/14)

[ch339]

Stacey posted the following problem:


* "For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by . If T is the sum of the first 10 terms in the sequence, then T is

"(A) greater than 2

"(B) between 1 and 2

"(C) between 1/2 and 1

"(D) between 1/4 and 1/2

"(E) less than 1/4"

Source: GMATPrep®

While I quickly figured out the pattern and solved, I am confused by the first "textbook" explanation (reproduced below). Why would we multiply the previous sum by the constant? I looked up the proof, which is similar to the "textbook solution," but it is not intuitive. Is there a good way to memorize the formula (the logic underlying the formula--I learn better that way)? Have you ever seen a series question that was hard to do by pattern recognition?

-------------------------------
When you have a geometric progression, you can calculate the sum in the following way:



Next, you’re going to multiply every term in the sum by the common ratio. What’s the common ratio? It’s the constant number that you keep multiplying each term by to get the next one. In this case, you’ve already figured this out: it’s - 1/2.

If you multiply this through all of the terms on both sides of the equation, you’ll get this:



Does anything look familiar? It’s basically the same list of numbers as in the first sum equation, except it’s missing the first number, 1/2. All of the others are identical!

Subtract this second equation from the first:



The right-hand side of the equation is always going to be just the first term of the original sum. The rest of the terms on the right-hand side of the two equations are identical, so when you subtract, they become zero and disappear.

Solve for s:

[RonPurewal]

It seems that there are some things missing from this post. Perhaps you wanted to embed something in it, but the embedding didn't work.

If this is on our site, you can just link to it; that would work. Alternatively, you could try posting this question as a comment on the blog post itself (which may be the best way, because the post's author will then reply to your comment).

[RonPurewal]

And this:

Have you ever seen a series question that was hard to do by pattern recognition?

Shouldn't ever happen.

The GMAT is designed so that memorization (other than memorization of extremely basic facts/fundamentals) affords the test taker as little advantage as possible. That's the entire purpose of the test"”it's just as "do-able" with only a basic knowledge of mathematics as it is with a math PhD.

If GMAC were to include a series-related problem that couldn't readily be solved by recognizing patterns, then that problem would fundamentally depend upon some "advanced" knowledge"”i.e., it would violate the entire spirit of the exam. So, I doubt it.

[ch339]

Hi Ron,

Thank you for your reassurance. The link to the problem is below.
Basically, it's a geometric series with a constant of -1/2. Can you provide some intuition behind the formula:

a * 1-r^n
---------------
1-r

where a is the first term, r is the constant, and n is the first n terms


Problem Given: Sum the first 10 terms, k is integer = [1, 10]

-1^ (k+1)
------------ --> 1/2, -1/4, 1/8 ...
2^k


http://www.manhattangmat.com/blog/index ... s-problem/

[RonPurewal]

Hi Ron,

Thank you for your reassurance. The link to the problem is below.
Basically, it's a geometric series with a constant of -1/2. Can you provide some intuition behind the formula:

a * 1-r^n
---------------
1-r

where a is the first term, r is the constant, and n is the first n terms

Heh, I am the wrong guy to ask about formulas "” there's no way I'd ever be able to remember a formula like that, unless I had to use it literally thousands of times (like, say, the Pythagorean theorem). In fact, it doesn't look very familiar.

But, I suppose you could derive it.

With the specifics you've provided, the sum of the sequence would be
S = a + ar + a(r^2) + ... + a(r^(n - 1))

If there is a formula like that, then it's clearly going to come from getting most of the terms to cancel out, in some way or another (since there's no "blah + blah + ... + blah" in the formula).
If we multiply the whole thing by r, then you have most of the same terms:
rS = ar + a(r^2) + ... + a(r^(n - 1)) + a(r^n)

Put them side by side, and notice all the terms that they have in common (purple):
S = a + ar + a(r^2) + ... + a(r^(n - 1))
rS = ar + a(r^2) + ... + a(r^(n - 1)) + a(r^n)

So, if you subtract them, you have
S - rS = a - a(r^(n - 1))

If you factor out "S" from the left side and then divide, you'll get this formula.

To me, not intuitive at all"”the formula results from multiplying by r (essentially at random) and then making a bunch of stuff cancel out. So, I'm not seeing any easy way to remember it.

Which, in turn, means that I wouldn't bother trying to remember it.
(:
(Honestly, it would probably just get in the way of pattern recognition, especially if I was given a sequence that didn't exactly fit the formula.)

[ch339]

Thanks, Ron! Makes a lot more sense now. And yeah, I will just keep trying to spot patterns.

[jlucero]

Glad it makes sense and let me second Ron's main point: algebra can get you an answer, but the GMAT isn't designed to test you on how much high school math you have memorized. It's designed to test you on how quickly you can recognize patterns and logical reasoning.