Numbers properties

[Xbond]

Could you explain how to resove this PS ?
See enclosed file

[khudania]

Id say E

[vivek.kapoor83]

IMO D. Wht is OA

[Xbond]

Guys,

I asked you what is the process to resolve this PS not the answser

regards,

[Stacey Koprince]

Hey, I received a PM asking me to comment on this one. Is it possible for you to type out the problem? If it contains a diagram you can't reproduce via text, then please embed an image file into a post rather than offering a file for download (if you want me to look at it). I don't download files anymore from the forums b/c of virus issues.

If you either type in the problem or embed the image file, send me another PM to let me know and I'll come and take a look!

[robzoc]

Here it´s is Stacey !

[Stacey Koprince]

This is a seriously hard problem. As always, make sure you glance at the answer choices before you start solving. See how they're grouped? So you don't need to find the exact number, just an approximation. And this is a function, so there's going to be some kind of pattern that lets us approximate because they just wouldn't make you calculate the sum of 10 terms in two minutes. (You can do that, of course - but there's got to be an easier way.)

So, figure out the first few terms and see if you can find the pattern. They tell me k is between 1 and 10, inclusive. So, if:
k=1, 1st term = 1*1/2 = 1/2
k = 2, 2nd term = (-1)*1/4 = -1/4
k = 3, 3rd term = 1*1/8 = 1/8

So look at how things are playing out. The first part of the function alternates between positive one (if the exponent is even) and negative one (if the exponent is odd). So all this will ever do is change the sign of the term.
The second part of the function keeps getting smaller and smaller: 1/2, 1/4, 1/8, ... And even here you can see the pattern - 2 keeps getting raised to the next power, so you could predict the rest of these, too.

So, you can do this next bit either just as math on paper, or you can draw a number line to visualize the swings back and forth.

So I start at 1/2 (my first term). Then I subtract 1/4, which gets me to 1/4. Then I add 1/8, which puts me halfway between 1/4 and 1/2. Then I subtract a number that is smaller than 1/8, so that puts me a little closer to 1/4 but still higher than 1/4. Then I add a number that's even smaller still, so that will put me higher than 1/4 but still not all the way up to or over 1/2. And so on. This is called converging - the number is converging on some number between 1/4 and 1/2, which is what choice D says.

I think this is easiest to see if you draw out a number line that's relatively to scale (which you can do, because the test gives you graph paper). So try that out and see if this converging thing makes sense to you!

[lunarpower]

as with most problems asking for the 10th (or 50th, or 408th, or ...) item in some sequence, the key to this one is to look for a pattern in the first few terms, and then extrapolate that pattern.

first term (= 'sum of first 1 terms') = +1/2
sum of first two terms = +1/2 - 1/4 = +1/4
sum of first three terms = +1/2 - 1/4 + 1/8 = +3/8
sum of first four terms = +1/2 - 1/4 + 1/8 - 1/16 = +5/16
sum of first five terms = ... + 1/32 = 11/32
... etc

notice that, from the sum of 3 terms onward, everything you'll get is between 1/4 and 1/2. once you see 3-4 repeats of this result, you can rest assured that the pattern will continue, so the answer is d.

---- nothing below this line is essential ----

theory corner:
you're adding / subtracting smaller and smaller values every time, which means you're moving in a smaller and smaller 'zigzag' around some eventually limiting value. (if you're a student in our course, the idea is markedly similar to the graphs we showed you when we talked about how the cat exam zeroes in on your ability level.) because of the diminishing value of the increments, results #3 and #4 above provide upper and lower bounds on all later values. because both of those values lie between 1/4 and 1/2, there's your answer.

theory corner #2:
if you know the formula for the sum of a geometric series (a/(1 - r)), you can figure out that the sum of the entire series (if it's continued out to infinity) is (1/2)/(1 + 1/2) = 1/3. that value is between 1/4 and 1/2; and, after ten rapidly diminishing terms, you know you're going to be pretty close to that.