For every integer k from 1 to 10

[Vinodbala]

Can someone explain how to solve the following problem? It appeared in one of the practice test.

For every integer k from 1 to 10, the kth term of a certain sequence is given by ((-1)^(k+1) ) * ((0.5)^k). If T is the sum of the first 10 terms in the sequence, the T is

a) greater than 2
b) between 1 and 2
c) between 0.5 and 1
d) between 0.25 and 0.5
e) less than 0.25

Correct answer is d

[sanjeev]

Hi,

Any term T(k) is given by (-1)^k+1 * (1/2)^k
Since we are asked to find out the sum of first 10 terms, we should find out how would the series would look like.

T(1) = (-1)^2 * (1/2)^1 = 1/2
T(2) = (-1)^3 * (1/2)^2 = -1/4
T(3) = (-1)^4 * (1/2)^3 = 1/8

Now, we can guess all the terms as we know every alternate term is multiple of -1/2 of previous term.
T(4) = -1/16


Let S(10) denote the sum of 10 such terms then,

S(10) = T(1) + T(2) + T(3) ..T(4).... T(10)

= (1/2 - 1/4) + (1/8 - 1/16) ....
= 1/4 + 1/16 + next values will be very insignificant ---- (1)
= .25 + .0625
= .31 + some very significant values

So Answere should be (d)


however if you want to get the exact solution,



From (1) we have, S(10) = 1/4 + 1/16 + 1/64 +... 5 terms in total(since every two terms gave us 1 term after subtracting)

So we have a Geometric progression with common ratio (1/4) , first term (1/4)

We know Sum of n terms in Geometric progression when common ratio (r) is < 1 is

S(n) = (a(1 - r^n)) / (1 -r )

for simplicity Taking (1/4) = .25

S(5) = .25(1 - (.25)^5)/ (1- .25)
= .25 (1 - (.25)^5) / .75
= (1 - (.25)^5)/3
NOw , the numerator 1 - (.25)^5 can be approximated to 1 since (.25)5 will be really small
S(5) = 1/3 = .33 approx

[RonPurewal]

here's another way to approach it.

this is an alternating series - i.e., a series of numbers in which the signs are + - + - + - ..., alternating every time. that is important in the discussion that follows.

the first 2 terms are 1/2 - 1/4. i will now prove that the solution is pinned down between those 2 values:

start with 1/2.
the next 2 terms are as follows: 1/2, minus something smaller, plus something even smaller than that. this must lead to something smaller than 1/2.
the same thing happens every following 2 terms, because the terms keep getting smaller and smaller. therefore, everything in the entire series will remain smaller than 1/2.

now start with 1/2 - 1/4 = 1/4 (the sum of the first 2 terms).
the next 2 terms are as follows: plus something smaller, minus something even smaller than that. this must lead to something bigger than 1/4.
the same thing happens every following 2 terms, because the terms keep getting smaller and smaller. therefore, everything in the entire series will remain bigger than 1/4.

taking these 2 observations together, we have that the final sum must be between 1/4 and 1/2.

--

of course, it's easier just to do pattern recognition: just find the sums for the first 4-5 examples, notice that ALL of them are between 0.25 and 0.5, and generalize accordingly. the gmat may be tricky about some things, but they are surprisingly foursquare about creating patterns that continue throughout the duration of a problem (i.e., they are unlikely to give you a problem in which the first twenty series are between 0.25 and 0.5 and then, surprise!).