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Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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math questions from GMAT PREP

by gana » Mon Jan 24, 2011 5:35 pm
For every integer k from 1 to 10 inclusive, the k th term of a certain sequence is given by (-1)^k+1 (1/2^k) if T is the sum of the first ten terms in the sequence, then T is .......?
a. between 1/2 and 3/4
b. between 1/4 and 1/2

Sorry guys I forgot to write all answer choice. but one of them is the correct answer. I would like to know how to solve it.

2. if it took Carlos 1/2 hour to cycle from his house to the library yesterday. Was the distance that he cycled yesterday greater than 6 miles. Note; 1 mile =5280 feet.
1. The average speed at which Carlos cycled from his house to teh library yesterday was greater than 16 feet per second.
2. The average speed at which Carlos cycled from his home to the library yesterday was less than 18 feet per second.

3. The perimeters of square region S AND rectangular region R are equal. If the sides of R are in the ratio of 2:3 , what is the ratio of area of the region R to teh area of the region S?
Last edited by gana on Tue Jan 25, 2011 12:57 pm, edited 1 time in total.
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by Tani » Tue Jan 25, 2011 12:11 pm
I don't understand the equation in the first question. What is the purpose of the 1 times (1/2)^2? Are you sure you copied it correctly? (-1)k will be -1 for the odd values of k (1,3,5,7,9) total -5 and +1 for the even values of k (2,4,6,8,10) for plus 5. Those cancel out and you are left with (1/2)^2 for each of the ten terms. That should be 10 times 1/4 or 2.5


At 1/2 hour, Carlos rode for 30*60 = 900 seconds.
6 miles would be 6*5280 = 31,680 feet.

Clue 1 = at more than 16 feet per second, Carlos went 16*900 or 14,480 - he could have gone 3 miles or 30 - insufficient

Clue 2 = at less than 18 feet per second Carlos went 18*900 feet or 16,200 feet. This is less than 6 miles and should be sufficient... unless...

It's not clear whether Carlos spent the night at the library or rode his bike back home and we don't know whether we are supposed to count his mileage in both directions. If we count both directions, clue 1 doesn't work and clue 2 does.

The third one is easiest if you pick numbers.

Assume the perimeter is 20. Each side of the square is 5 and the area is 25.

For the rectangle the perimeter is 2*length + 2* width = 20
therefore l+w = 10 and 3:2, one is 6 and the other 4.
The area of the rectangle is 4*6 = 24

R:S = 24:25
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by gana » Tue Jan 25, 2011 1:00 pm
Tani,

Sorry. Its my bad. I corrected the the first question. It was raised to the power of k not 2. Could you please look at them now?
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by gana » Tue Jan 25, 2011 1:08 pm
Hi Tani,
Thank you for your explanations. But I dont understand one more question from OG (green one, problem number 86). It says ; What is the largest integer n such that 1/2^n >0.01?
a.5
b.6
c.7
d.10
e.51.

OG explanation is that above equation is equivalent to 2^n<100. I dont know how 1/2^n > 0.01 is equivalent to 2^n<100. Is it rule or sth?
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by Tani » Tue Jan 25, 2011 1:56 pm
(1/2)^n is the same as (1^n)/(2^n), or 1/(2^n)

.01 is 1/100.

If you set them equal 1/(2^n) = 1/100 and cross multiply, you get 1 * 100 = 1 * (2^n) or 100 = 2^n

2^6 = 64
2^7 = 128

the answer is B
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by Tani » Tue Jan 25, 2011 2:11 pm
There is a formula for adding successive powers of a fraction, but i can't remember it, sorry. However, you can get close.

the powers of 1/2 are below

k=1 .5
k=2 .25
k=3 .125
k=4 .0625
k=5 .03125
k=6 .015625

(notice that each successive power is - naturally - half the one before it.)

etc.
after the first three you are at .875 so your answer will be more than 7/8
the additional pieces are getting very small so you are unlikely (even with this crude approach) to get to 1.00
therefore, I would look for an answer that puts the result between 9/10 and 1

If someone remembers the formula I would appreciate having it again.
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by gana » Tue Jan 25, 2011 6:03 pm
OG says between 1/4 and 1/2 for the first problem. I still dont get it.
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by Tani » Tue Jan 25, 2011 8:52 pm
can you send me the explanation?
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by gana » Tue Jan 25, 2011 11:16 pm
Hi Tani,

What kind of explanation are you asking ? Why I did not get your explanation or why the answer is between 1/4 and 1/2. The question is from Gmat prep CD, OR software. I took the practice test and I missed this question. There is no explanation how to solve it yet they show what is the right answer is. .
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by Tani » Wed Jan 26, 2011 9:59 pm
I thought you meant OG gave you an explanation. I get a different answer. Unless I've read the problem wrong, they are doing something I don't understand.
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by GMATGuruNY » Sun Jan 30, 2011 7:05 pm
For every integer k from 1 to 10,inclusive,the kth term of a certain sequence is given by (-1)^(k+1) * (1/2^k).If T is the sum of the 1st 10 terms in the sequence then T is,

a. > 2
b. between 1 and 2
c. between ½ and 1
d. between ¼ and ½
e. < ¼

OA D
Don't compute the exact sum.

If k=1, -1^(1+1)*(1/2*1) = 1/2
If k=2, -1^(2+1)*(1/2*2) = -1/4
Sum of the first two terms is 1/2 + ( -1/4) = 1/4.

If k=3, -1^(3+1)*(1/2*3) = 1/8.
If k=4, -1^(4+1)*(1/2*4) = -1/16

Now we can see the pattern.
The sum increases by a fraction (1/8, for example) and then decreases by a fraction 1/2 the size (-1/16, for example).
In other words, the sum will alternate between increasing a little and then decreasing a little less than it went up.

The sum of the first 2 terms is 1/4. Since all of the fractions after the first two terms will be less than 1/4, the sum will end up somewhere between 1/4 and 1/2.

The correct answer is D.
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