Sum between

[cesar.rodriguez.blanco]

What is 1/2+(1/2)^2+(1/2)^3+..............+(1/2)^20 between?

1/2 and 2/3
2/3 and 3/4
3/4 and 9/10
9/10 and 10/9
10/9 and 3/2

Source: gmatclub

[Kweku.Amoako]

You can use geometric series but that is the algebraic method.

a( first term) = 1/2
r (common ratio) = 1/2
n = number of terms = 20

I would apply some elimination technique....estimation

the value of a fraction with an integer exponent greater than 1 is smaller than the orignal fraction.

so (1/2) > (1/2)^2 because (1/2)^2 = 1/4
likewise (1/4) > (1/4)^2 because (1/4)^2 = 1/16

hence we know increasing powers of the fraction (1/2) exponentially decreases. So you can rewrite this expression as

1/2 + (less than 1/2 but greater than 0 ) + (less than a 4th but greater than 0) .....

just adding the first two terms 1/2 + 1/4 = 3/4 alone is greater than 2/3

A) since the the upper limit is 2/3 you can eliminate

B) If the first two terms add up 3/4 then the whole expression is definitely greater than 3/4

Recognize that 1/16 is small than 1/10 = 0.1 ...that means that powers of (1/2) greater > 4 will only contribute fractions of fractions of 0.1 so you can assume the sum will not add up to more than one since the 4th term and subsequent terms contribute nothing or very very small ( negligible values)

so based on the above I can eliminante D and E

C it is !!!!!( I HOPE ) made some rough assumptions ...let me know

[cesar.rodriguez.blanco]

Sorry, but the OA is D!!!!

[Ben Ku]

What is 1/2+(1/2)^2+(1/2)^3+..............+(1/2)^20 between?

1/2 and 2/3
2/3 and 3/4
3/4 and 9/10
9/10 and 10/9
10/9 and 3/2

Questions that involve sequences and series can be approached by discovering patterns or trends.

Let's add a few terms.
1/2 = 1/2
1/2 + (1/2)^2 = 1/2 + 1/4 = 3/4
1/2 + (1/2)^2 + (1/2)^3 = 3/4 + 1/8 = 7/8
1/2 + (1/2)^2 + (1/2)^3 + (1/2)^4 = 7/8 + 1/16 = 15/16

You might notice here that if you add up n terms, it will be (2^n - 1) / (2^n).
So if you add the first 20 terms, it'll be (2^20 - 1)/(2^20).

You might also notice that as you add each additional term, it gets you closer to 1, but never goes above 1. So the sum will be somewhere close to, but less than 1. The only answer choice that works is (D).

[Kweku.Amoako]

tx Ben,

would you also advice the estimation method on this question? I made the assumption that the sum < 0 but apparently did not get the right answer

Also if the as assumption is valid, why is D right since the upper limit in D is greater than 1.

[Ben Ku]

would you also advice the estimation method on this question? I made the assumption that the sum < 0 but apparently did not get the right answer

Also if the as assumption is valid, why is D right since the upper limit in D is greater than 1.

Your approach was not bad; you just didn't go far enough. Be careful what kinds of assumptions you make. When working on such problems, LOOK FOR PATTERNS.

The answer choice just says that the sum is BETWEEN 9/10 and 10/9. Just because the upper limit is greater than 1 doesn't mean the sum needs to be.