(a) Prove the result in the earlier example by induction.

(b) Prove the result of the earlier example directly by recursively substituting (1+xk)=(1x2k)/(1xk)(1+x^k )= (1−x^{2k} )/(1−x^k) in g*(x).


Answered 1 year ago
Answered 1 year ago
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a) To prove the result, we will use strong induction since during the induction step, we will need to rely on the results for several smaller numbers than the current one.

Start with the induction basis, i.e. the smallest natural number 11 which can obviously be represented uniquely as a sum of powers of 22 since 1=201=2^0.

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