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 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
This is a question from GMAT Prep, as I checked the result, the right answer is D, but I have no idea how to approach this question.
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lunarpower
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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.
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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.
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.
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Voit esittää kysymyksiä Ron:lle myös suomeksi
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Anurag@Gurome
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Refer to the post: https://www.beatthegmat.com/then-t-is-gm ... 15193.htmldtl wrote: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 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
This is a question from GMAT Prep, as I checked the result, the right answer is D, but I have no idea how to approach this question.
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GMATGuruNY
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Calculate until you see the pattern.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 first 10 terms in the sequence, then T is
A. Greater than 2
B. Between 1 and 2
C. Between 1/2 to 1
D. Between 1/4 to1/2
E. Less than ¼
Some test-takers might find it helpful to visualize the sum on a number line.
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).
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. From there, the sum will increase by 1/8, decrease by a smaller fraction (1/16), increase by an even smaller fraction (1/32), and so on. Here are the first four terms, plotted on a number line:
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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BTG14
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Nice approach GMAT guru..
Still I went with formula approach.
As GMAT guru explained it's a G.P(Geometric Progression), a=1/2, r=-1/2, n=10
Sum of First 10 terms= a (1-r^n)/(1-r) = 1/2(1-1/2^10)/(1-r)= 1/2*2/3(1-1/2^10) = 1/3(1-1/2^10).
Since 1-1/2^10 is almost equal to 1 we can assume answer will be near 1/3 , which is in between 1/4 and 1/2.
Still I went with formula approach.
As GMAT guru explained it's a G.P(Geometric Progression), a=1/2, r=-1/2, n=10
Sum of First 10 terms= a (1-r^n)/(1-r) = 1/2(1-1/2^10)/(1-r)= 1/2*2/3(1-1/2^10) = 1/3(1-1/2^10).
Since 1-1/2^10 is almost equal to 1 we can assume answer will be near 1/3 , which is in between 1/4 and 1/2.
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varun289
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can i say a short cut -
Ist term = 1/2
secodn = -1/(2^2)
III= 1/2^3
IV = -1/2^4
pattern= + - + - with 1/2
so add forst two term only = 1/2+(-1/2^2) = 1/2^2 , same will goes on
and you get ans = 1/4 + bal bla bla = so range will be 1/4 to 1/2
simply cooooolllll
Ist term = 1/2
secodn = -1/(2^2)
III= 1/2^3
IV = -1/2^4
pattern= + - + - with 1/2
so add forst two term only = 1/2+(-1/2^2) = 1/2^2 , same will goes on
and you get ans = 1/4 + bal bla bla = so range will be 1/4 to 1/2
simply cooooolllll
