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18 Jul 2009, 10:00
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18 Jul 2009, 13:01
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I got this questions while doing the simulation as well.
This is what I did.
The first part of the equation, \((-1)^k+1\), will only determine whether the number is positive or negative.
For k = 1, this part will be 1
For k = 2, this part will be -1
For k = 3, this part will be 1, again. Thus, when k is even the term will be multiple by -1, and it will be negative. However, when k is odd the term will be positive.
The second part of the equation just tell us that is a fraction of power of 2.
Thus the numbers are:
\(1/2, -1/4, 1/8, -1/16, 1/32, -1/64, 1/128, -1/256, 1/512, -1/1024\)
Now try to sum the first positives to have a impression of the sum of the positives, and do the same with the negatives.
If you transform them to decimal, you will notice that the first positives, are, 0.5, 0.125, 0.03, now hold on. The sum here is 0.655. The way the numbers are going down to fast, you can see that this sum will not pass 0.7, or if it pass, it will be very close to 0.7. Save this number.
Now, if you sum the negatives, in decimal, you will see -0.25, -0.06, ok that is enough. You do not need even to continue. As you can see, again the numbers are going down ("modularity talking") so fast that you can see the the sum will be around -0.3.
Thus \(0.7 - 0.3 =~ 0.4\), your answer approximately.
D between 0.25 and 0.5
This is what I did.
The first part of the equation, \((-1)^k+1\), will only determine whether the number is positive or negative.
For k = 1, this part will be 1
For k = 2, this part will be -1
For k = 3, this part will be 1, again. Thus, when k is even the term will be multiple by -1, and it will be negative. However, when k is odd the term will be positive.
The second part of the equation just tell us that is a fraction of power of 2.
Thus the numbers are:
\(1/2, -1/4, 1/8, -1/16, 1/32, -1/64, 1/128, -1/256, 1/512, -1/1024\)
Now try to sum the first positives to have a impression of the sum of the positives, and do the same with the negatives.
If you transform them to decimal, you will notice that the first positives, are, 0.5, 0.125, 0.03, now hold on. The sum here is 0.655. The way the numbers are going down to fast, you can see that this sum will not pass 0.7, or if it pass, it will be very close to 0.7. Save this number.
Now, if you sum the negatives, in decimal, you will see -0.25, -0.06, ok that is enough. You do not need even to continue. As you can see, again the numbers are going down ("modularity talking") so fast that you can see the the sum will be around -0.3.
Thus \(0.7 - 0.3 =~ 0.4\), your answer approximately.
D between 0.25 and 0.5
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Problem Solving (PS) Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
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