Prove that for any positive integer n, there is a k e Z2o and ao, a1,..., apk € {0,1} such that k n = i=0 k (For those of you familiar with binary, the equality n = E a;2' is the same as saying i=0 that n is given by akak-1·.. a1ao 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 E Z = 2m +1 for some m E Z. You may use this fact without proof if you'd like.) or n =
Prove that for any positive integer n, there is a k e Z2o and ao, a1,..., apk € {0,1} such that k n = i=0 k (For those of you familiar with binary, the equality n = E a;2' is the same as saying i=0 that n is given by akak-1·.. a1ao 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 E Z = 2m +1 for some m E Z. You may use this fact without proof if you'd like.) or n =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba208968-0ccc-4a94-8921-5ea260cefe19%2Ff638e432-c721-4f16-99ff-0ae3a84610c6%2Fr0qu0rl_processed.png&w=3840&q=75)
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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