Prove by induction binary base
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Prove by induction that every natural number $n$ has a natural number $k$ and numbers $a_{0} , a_{1},..., a_{k}in left { 0,1 right }$ such that $n=sum_{i=0}^{k} a_{i} 2^{i}$
I really don't understand What should I do here? I guess it's related to a binary base but how to start solving this question?
Thanks!
induction
edited Nov 22 at 16:13
user3482749
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asked Nov 22 at 16:11
Marry G
32
add a comment |
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Prove by induction that every natural number $n$ has a natural number $k$ and numbers $a_{0} , a_{1},..., a_{k}in left { 0,1 right }$ such that $n=sum_{i=0}^{k} a_{i} 2^{i}$
I really don't understand What should I do here? I guess it's related to a binary base but how to start solving this question?
Thanks!
induction
edited Nov 22 at 16:13
user3482749
2,166414
asked Nov 22 at 16:11
Marry G
32
Do induction on n. The base case is trivial.
– GNUSupporter 8964民主女神 地下教會
Nov 22 at 16:30
add a comment |
up vote
-2
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up vote
-2
down vote
favorite
Prove by induction that every natural number $n$ has a natural number $k$ and numbers $a_{0} , a_{1},..., a_{k}in left { 0,1 right }$ such that $n=sum_{i=0}^{k} a_{i} 2^{i}$
I really don't understand What should I do here? I guess it's related to a binary base but how to start solving this question?
Thanks!
induction
edited Nov 22 at 16:13
user3482749
2,166414
asked Nov 22 at 16:11
Marry G
32
Prove by induction that every natural number $n$ has a natural number $k$ and numbers $a_{0} , a_{1},..., a_{k}in left { 0,1 right }$ such that $n=sum_{i=0}^{k} a_{i} 2^{i}$
I really don't understand What should I do here? I guess it's related to a binary base but how to start solving this question?
Thanks!
induction
induction
edited Nov 22 at 16:13
user3482749
2,166414
asked Nov 22 at 16:11
Marry G
32
edited Nov 22 at 16:13
user3482749
2,166414
asked Nov 22 at 16:11
Marry G
32
edited Nov 22 at 16:13
user3482749
2,166414
edited Nov 22 at 16:13
user3482749
2,166414
edited Nov 22 at 16:13
user3482749
2,166414
2,166414
asked Nov 22 at 16:11
Marry G
32
asked Nov 22 at 16:11
Marry G
32
asked Nov 22 at 16:11
Marry G
32
32
Do induction on n. The base case is trivial.
– GNUSupporter 8964民主女神 地下教會
Nov 22 at 16:30
add a comment |
Do induction on n. The base case is trivial.
– GNUSupporter 8964民主女神 地下教會
Nov 22 at 16:30
Do induction on n. The base case is trivial.
– GNUSupporter 8964民主女神 地下教會
Nov 22 at 16:30
Do induction on n. The base case is trivial.
– GNUSupporter 8964民主女神 地下教會
Nov 22 at 16:30
add a comment |
1 Answer
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You've correctly detected that this is connected with the binary expression for $n$. You need to do three things now: (1) Figure out exactly what that connection is, i.e., if I gave you the binary expression for $n$, say 110101 when n=53, how could you easily find the $a_i$'s? (2) Prove the base case of the desired induction, i.e., prove that the result is correct for $n=0$ (or $n=1$ if your natural numbers begin with $1$). (3) Prove the induction step, i.e., assuming the desired result for $n$ prove it for $n+1$. In part (3), it will be useful to understand exactly how adding $1$ works in binary notation, in particular what "carrying" occurs. Once you understand that in terms of binary notation, you can use (1) to express it in terms of the $a_i$'s and so complete the induction proof.
answered Nov 22 at 16:31
Andreas Blass
48.9k350106
Thank you very much, excellent explanation!
– Marry G
Nov 22 at 16:47
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
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votes
up vote
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down vote
accepted
You've correctly detected that this is connected with the binary expression for $n$. You need to do three things now: (1) Figure out exactly what that connection is, i.e., if I gave you the binary expression for $n$, say 110101 when n=53, how could you easily find the $a_i$'s? (2) Prove the base case of the desired induction, i.e., prove that the result is correct for $n=0$ (or $n=1$ if your natural numbers begin with $1$). (3) Prove the induction step, i.e., assuming the desired result for $n$ prove it for $n+1$. In part (3), it will be useful to understand exactly how adding $1$ works in binary notation, in particular what "carrying" occurs. Once you understand that in terms of binary notation, you can use (1) to express it in terms of the $a_i$'s and so complete the induction proof.
answered Nov 22 at 16:31
Andreas Blass
48.9k350106
Thank you very much, excellent explanation!
– Marry G
Nov 22 at 16:47
add a comment |
up vote
0
down vote
accepted
You've correctly detected that this is connected with the binary expression for $n$. You need to do three things now: (1) Figure out exactly what that connection is, i.e., if I gave you the binary expression for $n$, say 110101 when n=53, how could you easily find the $a_i$'s? (2) Prove the base case of the desired induction, i.e., prove that the result is correct for $n=0$ (or $n=1$ if your natural numbers begin with $1$). (3) Prove the induction step, i.e., assuming the desired result for $n$ prove it for $n+1$. In part (3), it will be useful to understand exactly how adding $1$ works in binary notation, in particular what "carrying" occurs. Once you understand that in terms of binary notation, you can use (1) to express it in terms of the $a_i$'s and so complete the induction proof.
answered Nov 22 at 16:31
Andreas Blass
48.9k350106
Thank you very much, excellent explanation!
– Marry G
Nov 22 at 16:47
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
You've correctly detected that this is connected with the binary expression for $n$. You need to do three things now: (1) Figure out exactly what that connection is, i.e., if I gave you the binary expression for $n$, say 110101 when n=53, how could you easily find the $a_i$'s? (2) Prove the base case of the desired induction, i.e., prove that the result is correct for $n=0$ (or $n=1$ if your natural numbers begin with $1$). (3) Prove the induction step, i.e., assuming the desired result for $n$ prove it for $n+1$. In part (3), it will be useful to understand exactly how adding $1$ works in binary notation, in particular what "carrying" occurs. Once you understand that in terms of binary notation, you can use (1) to express it in terms of the $a_i$'s and so complete the induction proof.
answered Nov 22 at 16:31
Andreas Blass
48.9k350106
You've correctly detected that this is connected with the binary expression for $n$. You need to do three things now: (1) Figure out exactly what that connection is, i.e., if I gave you the binary expression for $n$, say 110101 when n=53, how could you easily find the $a_i$'s? (2) Prove the base case of the desired induction, i.e., prove that the result is correct for $n=0$ (or $n=1$ if your natural numbers begin with $1$). (3) Prove the induction step, i.e., assuming the desired result for $n$ prove it for $n+1$. In part (3), it will be useful to understand exactly how adding $1$ works in binary notation, in particular what "carrying" occurs. Once you understand that in terms of binary notation, you can use (1) to express it in terms of the $a_i$'s and so complete the induction proof.
answered Nov 22 at 16:31
Andreas Blass
48.9k350106
answered Nov 22 at 16:31
Andreas Blass
48.9k350106
answered Nov 22 at 16:31
Andreas Blass
48.9k350106
answered Nov 22 at 16:31
Andreas Blass
48.9k350106
48.9k350106
Thank you very much, excellent explanation!
– Marry G
Nov 22 at 16:47
add a comment |
Thank you very much, excellent explanation!
– Marry G
Nov 22 at 16:47
Thank you very much, excellent explanation!
– Marry G
Nov 22 at 16:47
Thank you very much, excellent explanation!
– Marry G
Nov 22 at 16:47
add a comment |
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Do induction on n. The base case is trivial.
– GNUSupporter 8964民主女神 地下教會
Nov 22 at 16:30