Proof Ideas - Strong Induction, Pascal's Triangle and Fibonacci Numbers
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I'm looking for attributes/characteristics/properties to prove about Pascal's Triangle or the Fibonacci numbers. Preferably something that requires a strong induction proof that is on the same level as proving things such as the sum of the elements in the nth row of Pascal's triangle is $2^n$. Simple induction proof ideas are also fine.
reference-request induction binomial-coefficients fibonacci-numbers
edited Dec 5 '18 at 22:58
Scientifica
6,37641335
asked Dec 5 '18 at 22:37
DanielleDanielle
2179
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add a comment |
$begingroup$
I'm looking for attributes/characteristics/properties to prove about Pascal's Triangle or the Fibonacci numbers. Preferably something that requires a strong induction proof that is on the same level as proving things such as the sum of the elements in the nth row of Pascal's triangle is $2^n$. Simple induction proof ideas are also fine.
reference-request induction binomial-coefficients fibonacci-numbers
edited Dec 5 '18 at 22:58
Scientifica
6,37641335
asked Dec 5 '18 at 22:37
DanielleDanielle
2179
$endgroup$
1
$begingroup$
How about Pascal's triangle and the Fibonacci numbers? Try proving that $F_{n+1} = sum_{k ge 0} {n-k choose k}$.
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– Qiaochu Yuan
Dec 5 '18 at 23:45
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The known properties of Pascal's triangle, and of the Fibonacci sequence, that can be proved by induction, can each fill a library.
$endgroup$
– DanielWainfleet
Dec 6 '18 at 9:11
add a comment |
$begingroup$
I'm looking for attributes/characteristics/properties to prove about Pascal's Triangle or the Fibonacci numbers. Preferably something that requires a strong induction proof that is on the same level as proving things such as the sum of the elements in the nth row of Pascal's triangle is $2^n$. Simple induction proof ideas are also fine.
reference-request induction binomial-coefficients fibonacci-numbers
edited Dec 5 '18 at 22:58
Scientifica
6,37641335
asked Dec 5 '18 at 22:37
DanielleDanielle
2179
$endgroup$
I'm looking for attributes/characteristics/properties to prove about Pascal's Triangle or the Fibonacci numbers. Preferably something that requires a strong induction proof that is on the same level as proving things such as the sum of the elements in the nth row of Pascal's triangle is $2^n$. Simple induction proof ideas are also fine.
reference-request induction binomial-coefficients fibonacci-numbers
reference-request induction binomial-coefficients fibonacci-numbers
edited Dec 5 '18 at 22:58
Scientifica
6,37641335
asked Dec 5 '18 at 22:37
DanielleDanielle
2179
edited Dec 5 '18 at 22:58
Scientifica
6,37641335
asked Dec 5 '18 at 22:37
DanielleDanielle
2179
edited Dec 5 '18 at 22:58
Scientifica
6,37641335
edited Dec 5 '18 at 22:58
Scientifica
6,37641335
edited Dec 5 '18 at 22:58
Scientifica
6,37641335
6,37641335
asked Dec 5 '18 at 22:37
DanielleDanielle
2179
asked Dec 5 '18 at 22:37
DanielleDanielle
2179
asked Dec 5 '18 at 22:37
DanielleDanielle
2179
2179
1
$begingroup$
How about Pascal's triangle and the Fibonacci numbers? Try proving that $F_{n+1} = sum_{k ge 0} {n-k choose k}$.
$endgroup$
– Qiaochu Yuan
Dec 5 '18 at 23:45
$begingroup$
The known properties of Pascal's triangle, and of the Fibonacci sequence, that can be proved by induction, can each fill a library.
$endgroup$
– DanielWainfleet
Dec 6 '18 at 9:11
add a comment |
1
$begingroup$
How about Pascal's triangle and the Fibonacci numbers? Try proving that $F_{n+1} = sum_{k ge 0} {n-k choose k}$.
$endgroup$
– Qiaochu Yuan
Dec 5 '18 at 23:45
$begingroup$
The known properties of Pascal's triangle, and of the Fibonacci sequence, that can be proved by induction, can each fill a library.
$endgroup$
– DanielWainfleet
Dec 6 '18 at 9:11
1
1
$begingroup$
How about Pascal's triangle and the Fibonacci numbers? Try proving that $F_{n+1} = sum_{k ge 0} {n-k choose k}$.
$endgroup$
– Qiaochu Yuan
Dec 5 '18 at 23:45
$begingroup$
How about Pascal's triangle and the Fibonacci numbers? Try proving that $F_{n+1} = sum_{k ge 0} {n-k choose k}$.
$endgroup$
– Qiaochu Yuan
Dec 5 '18 at 23:45
$begingroup$
The known properties of Pascal's triangle, and of the Fibonacci sequence, that can be proved by induction, can each fill a library.
$endgroup$
– DanielWainfleet
Dec 6 '18 at 9:11
$begingroup$
The known properties of Pascal's triangle, and of the Fibonacci sequence, that can be proved by induction, can each fill a library.
$endgroup$
– DanielWainfleet
Dec 6 '18 at 9:11
add a comment |
0
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$begingroup$
How about Pascal's triangle and the Fibonacci numbers? Try proving that $F_{n+1} = sum_{k ge 0} {n-k choose k}$.
$endgroup$
– Qiaochu Yuan
Dec 5 '18 at 23:45
$begingroup$
The known properties of Pascal's triangle, and of the Fibonacci sequence, that can be proved by induction, can each fill a library.
$endgroup$
– DanielWainfleet
Dec 6 '18 at 9:11