Re: The value of (2^(-14) + 2^(-15) + 2^(-16) + 2^(-17))/5 is
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18 May 2022, 12:16
I solved this a slightly different way (more based on reasoning);
To start, I noted that each of the terms in the numerator can be broken out separately, giving us one of the operations as 2^-17 / 5 (remembering that a + b / c = a / c + b / c).
In comparing this to the value the problem is asking us to evaluation, we can note that this specific operation (mentioned above) will represent 1/5 of the value of 2^-17.
Moving down the line, we come across our second operation 2^-16 / 5. Thinking logically, I can presume that there is one less (1/2) in this expression than in the preceding operation (2^-17 / 5) since 2^-16 can also be written as 1/2^16.
Thus, in order to compare magnitudes from the first operation we evaluated, we can multiply the original magnitude by 2, yielding us 2 / 5 of the value the problem is asking us to evaluate (1 / 5 * 2 = 2 / 5) (Note: We multiply by 2 here because for our term, 2^-16, to have one less (1/2) means that there was a (1/2) divided out of the original operation, 2^-17. Therefore, to divide out a (1/2) is the equivalent of multiplying our operation, 2^-17 by 2 to yield 2^-16).
If we continue this train of thought for each term, we'll recognize 2^-15 has TWO less (1/2)'s, therefore we multiply our magnitude by 4... and so on and so forth.
Adding up all of our terms' magnitudes in relation to the value the problem is asking us to evaluate, we end with 15 / 5 or 3.
I will admit, algebra would have been easier, but I like to think through these things.
Feel free to correct me if my logic is flawed in any of the above.