Problem Solving

Some of these problems are driving me nuts. I am missing something for sure. Any help would be much appreciated. Thanks!

For this one, I obvious can do it the long way, but it is not feasible at test time. What is the best method to solve this problem?

For every integer k from 10 to 10 inclusive, the kth term is given by ((-1)^(k+1))*(1/(2^k)). What is the sum of the first 10 terms?

Answer: between 1/4 and 1/2

I get:
k=1, term = 1/2
k=2, term = -1/4
k=3, term = 1/16
k=4, term = -1/32
etc.
Summing these numbers for K=1 through 10 is for sure not the best way to solve this problem. What is?

RachelAL wrote:For every integer k from 10 to 10 inclusive, the kth term is given by ((-1)^(k+1))*(1/(2^k)). What is the sum of the first 10 terms?

Are you sure you typed the problem right?
Sorry I could really get what the problem is all about... but then source is GMAT Prep so I really cant doubt about the question it has to be right..

Did you mean : "For every integer k from 0 to 10 inclusive,"
Or is it from 1 to 10? because that could change the answer !!!
Also please provide all options given.
hope you have gone through https://www.beatthegmat.com/viewtopic.php?t=1133
Anyways back to the problem:

RachelAL wrote: For this one, I obvious can do it the long way, but it is not feasible at test time. What is the best method to solve this problem?

For every integer k from 10 to 10 inclusive, the kth term is given by ((-1)^(k+1))*(1/(2^k)). What is the sum of the first 10 terms?

Answer: between 1/4 and 1/2

I get:
k=1, term = 1/2
k=2, term = -1/4
k=3, term = 1/16
k=4, term = -1/32
etc.
Summing these numbers for K=1 through 10 is for sure not the best way to solve this problem. What is?

See you got the trend right!!
alternate terms are negative and one more trend you must have observed that each higher number term has less and less value and that too in 2^n order... so you dont really have to calculate all this...

I hope you understand the trick here.
Always remember if you see such nuts surely there's some easy way out some short cuts *(I call it trend : you can search for this word & you 'll find similar such posts)

It is 1 to 10.

I see I got the trend right, but do I still have to add all 10 terms? Once I get the 10 terms, I think I a missing the shortcut to add them.

Thanks for your response.

they seem easy to add once you get the trend.
but you dont even have to add them till 10th term once you get the trend...
it is clear that the value of sum is settling down between 0.25 & 0.5

The terms 1/2, -1/4, 1/8, -1/16 constitute a geometric progression with first term 1/2 and a common ratio of -1/2

The sum of a geometric progression is given by
Sum= a{(r^n - 1)/(r-1)} when r>1
= a{(1 - r^n)/(1-r)} when r<1
where a is the first term, r is the common ration and n is the total number of terms.

Using this formula;
a=1/2
r= -1/2
n=10

I am skipping some arithmetic steps

in the end u get (1/3) * (1023/1024)
{here, it is imp to know the values of powers of 2 upto 10 : 2^10 = 1024}

1023/1024 is approx 1

therefore answer is approx 1/3 = 0.33 which is between 0.25 and 0.50