AMC 10/12 A 2020 (Problem 21)
There exists a unique strictly increasing sequence of nonnegative integers such that . What is ?
Use the identity for the sum of odd powers (when the exponent is odd): . Here and is odd.
Factor the numerator: . Therefore, . This is an alternating sum of 17 terms, starting with positive and ending with .
Key fact we'll reuse: . This lets us rewrite differences like as a sum of distinct powers of with consecutive exponents.
Look at one “block”: . Now expand the difference of powers in base 2: . More cleanly (using geometric series structure): . This is a sum of 16 consecutive powers: Including also that final lone from elsewhere will give us 17 distinct powers in total for that pair. In words: each adjacent pair “unpacks” into 17 distinct powers of 2 with consecutive exponents.
Similarly, for higher exponents: . Again we get 17 distinct consecutive powers of 2.
The whole expression can be grouped into 8 “difference blocks” of the form , each block expanding into 17 distinct powers of with unique exponents, plus the final . Because the exponents in different blocks don't overlap, the binary expansion of the entire sum is just a bunch of distinct powers of 2, no carries needed. So we get: - one term from the final , - plus blocks, - with distinct powers of contributed per block.
Therefore .
(C) 137