Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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can someone please answer this in detail? I am very confused so please show all the steps.
![Prove that for any positive integer \( n \), there is a \( k \in \mathbb{Z}_{\geq 0} \) and \( a_0, a_1, \ldots, a_k \in \{0,1\} \) such that
\[
n = \sum_{i=0}^{k} a_i 2^i.
\]
(For those of you familiar with binary, the equality \( n = \sum_{i=0}^{k} a_i 2^i \) is the same as saying that \( n \) is given by \( a_k a_{k-1} \cdots a_1 a_0 \) in binary. There are several ways to prove this, but one of them uses the fact that given a positive integer \( n \), either \( n = 2m \) for some \( m \in \mathbb{Z} \) or \( n = 2m + 1 \) for some \( m \in \mathbb{Z} \). You may use this fact without proof if you’d like.)](https://content.bartleby.com/qna-images/question/ba208968-0ccc-4a94-8921-5ea260cefe19/f638e432-c721-4f16-99ff-0ae3a84610c6/r0qu0rl_processed.png)
Transcribed Image Text:Prove that for any positive integer \( n \), there is a \( k \in \mathbb{Z}_{\geq 0} \) and \( a_0, a_1, \ldots, a_k \in \{0,1\} \) such that
\[
n = \sum_{i=0}^{k} a_i 2^i.
\]
(For those of you familiar with binary, the equality \( n = \sum_{i=0}^{k} a_i 2^i \) is the same as saying that \( n \) is given by \( a_k a_{k-1} \cdots a_1 a_0 \) in binary. There are several ways to prove this, but one of them uses the fact that given a positive integer \( n \), either \( n = 2m \) for some \( m \in \mathbb{Z} \) or \( n = 2m + 1 \) for some \( m \in \mathbb{Z} \). You may use this fact without proof if you’d like.)
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