Simplify this expression:
2^23 - 2^22 - 2^21 - 1
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1 + 2 + 2^2 + ... + 2^20
We can generalize this a bit, into the expression:
2^a - 2^(a-1) - ... - 2^(b+2) - 2^(b+1) - 1
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1 + 2 + 2^2 + ... + 2^(b-1) + 2^b
Then this problem is the generalization at a=23 and b=20.
The replace every negated power in the numerator into a difference: - 2^k = 2^k - 2^(k+1).
Then the numerator becomes 2^a + [2^(a-1) - 2^a] + [2^(a-2) - 2^(a-1)] + ... + [2^(b+2) - 2^(b+3)] + [2^(b+1) - 2^(b+2)] - 1
In this form it is easy to see the numerator is now a telescopic series and collapses into the simplified form of 2^(b+1) - 1.
The denominator can be handled similarly and will also have a simplified form of 2^(b+1) - 1.
Since the numerator and denominator are identical then the generalized expression will reduce to 1 for any choice of a and b.
Solution
