Show that $f_n+1cdot f_n-1-f^2_n=(-1)^n$ if $f_n $ is the $n$th Fibonacci number The Next CEO of Stack OverflowFibonacci proof question: $f_n+1f_n-1-f_n^2=(-1)^n$Fibonacci sequence: how to prove that $alpha^n=alphacdot F_n + F_n-1$?Prove the Number of Additions of Fibonacci Number AlgorithmInduction, show that something is smaller then …Proving this $F_n+1 cdot F_n-1 - F^2_n = (-1)^n$ by inductionProving $F_n ge (frac12(1+sqrt5))^n-2$ for $n in mathbbN_>1$ when $F_n$ is the nth Fibonacci numberShow by induction that $n! > n cdot F_n$ for all $n > 3$Show $F_n+1 cdot F_n-1 = F_n^2 + (-1)^n$ for all $n in mathbbN$$gcd(f_k, f_k+3)$, where $f_k$ is the k'th Fibonacci numberShow $f_2n = f_n^2 + 2f_n-1f_n$ where $f_n$ is the $n^th$ fibonacci numberExplaining the proof of Fibonacci number using inductive reasoning
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Show that $f_n+1cdot f_n-1-f^2_n=(-1)^n$ if $f_n $ is the $n$th Fibonacci number
The Next CEO of Stack OverflowFibonacci proof question: $f_n+1f_n-1-f_n^2=(-1)^n$Fibonacci sequence: how to prove that $alpha^n=alphacdot F_n + F_n-1$?Prove the Number of Additions of Fibonacci Number AlgorithmInduction, show that something is smaller then …Proving this $F_n+1 cdot F_n-1 - F^2_n = (-1)^n$ by inductionProving $F_n ge (frac12(1+sqrt5))^n-2$ for $n in mathbbN_>1$ when $F_n$ is the nth Fibonacci numberShow by induction that $n! > n cdot F_n$ for all $n > 3$Show $F_n+1 cdot F_n-1 = F_n^2 + (-1)^n$ for all $n in mathbbN$$gcd(f_k, f_k+3)$, where $f_k$ is the k'th Fibonacci numberShow $f_2n = f_n^2 + 2f_n-1f_n$ where $f_n$ is the $n^th$ fibonacci numberExplaining the proof of Fibonacci number using inductive reasoning
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I've reviewed this answer: https://math.stackexchange.com/a/606286/584468 and I'm getting lost on how he did $f_k+2f_k−f^2_k+1=(f_k−f_k+1)f_k−f^2_k+1$
When I thought that $f_k+2=f_k+1+f_k$?
But I'm getting:
$$(f_k+1+f_k)cdot f_k-f^2_k+1 = f_k+1cdot f_k +f^2_k - f^2_k+1$$
$$=(f_k+1)(f_k-f_k+1) + f^2_k$$
$$=(f_k+1)(f_k-(f_k+f_k-1)) + f^2_k $$
$$=f_k+1cdot(-f_k-1) + f^2_k$$
$$=-f_k+1cdot f_k-1 + f^2_k$$
$$=(-1)(f_k+1cdot f_k-1 - f^2_k)$$
$$=(-1)(-1)^k$$
induction recurrence-relations fibonacci-numbers
asked Mar 28 at 2:40
ElliottElliott
947
$endgroup$
add a comment |
$begingroup$
I've reviewed this answer: https://math.stackexchange.com/a/606286/584468 and I'm getting lost on how he did $f_k+2f_k−f^2_k+1=(f_k−f_k+1)f_k−f^2_k+1$
When I thought that $f_k+2=f_k+1+f_k$?
But I'm getting:
$$(f_k+1+f_k)cdot f_k-f^2_k+1 = f_k+1cdot f_k +f^2_k - f^2_k+1$$
$$=(f_k+1)(f_k-f_k+1) + f^2_k$$
$$=(f_k+1)(f_k-(f_k+f_k-1)) + f^2_k $$
$$=f_k+1cdot(-f_k-1) + f^2_k$$
$$=-f_k+1cdot f_k-1 + f^2_k$$
$$=(-1)(f_k+1cdot f_k-1 - f^2_k)$$
$$=(-1)(-1)^k$$
induction recurrence-relations fibonacci-numbers
asked Mar 28 at 2:40
ElliottElliott
947
$endgroup$
add a comment |
$begingroup$
I've reviewed this answer: https://math.stackexchange.com/a/606286/584468 and I'm getting lost on how he did $f_k+2f_k−f^2_k+1=(f_k−f_k+1)f_k−f^2_k+1$
When I thought that $f_k+2=f_k+1+f_k$?
But I'm getting:
$$(f_k+1+f_k)cdot f_k-f^2_k+1 = f_k+1cdot f_k +f^2_k - f^2_k+1$$
$$=(f_k+1)(f_k-f_k+1) + f^2_k$$
$$=(f_k+1)(f_k-(f_k+f_k-1)) + f^2_k $$
$$=f_k+1cdot(-f_k-1) + f^2_k$$
$$=-f_k+1cdot f_k-1 + f^2_k$$
$$=(-1)(f_k+1cdot f_k-1 - f^2_k)$$
$$=(-1)(-1)^k$$
induction recurrence-relations fibonacci-numbers
asked Mar 28 at 2:40
ElliottElliott
947
$endgroup$
I've reviewed this answer: https://math.stackexchange.com/a/606286/584468 and I'm getting lost on how he did $f_k+2f_k−f^2_k+1=(f_k−f_k+1)f_k−f^2_k+1$
When I thought that $f_k+2=f_k+1+f_k$?
But I'm getting:
$$(f_k+1+f_k)cdot f_k-f^2_k+1 = f_k+1cdot f_k +f^2_k - f^2_k+1$$
$$=(f_k+1)(f_k-f_k+1) + f^2_k$$
$$=(f_k+1)(f_k-(f_k+f_k-1)) + f^2_k $$
$$=f_k+1cdot(-f_k-1) + f^2_k$$
$$=-f_k+1cdot f_k-1 + f^2_k$$
$$=(-1)(f_k+1cdot f_k-1 - f^2_k)$$
$$=(-1)(-1)^k$$
induction recurrence-relations fibonacci-numbers
induction recurrence-relations fibonacci-numbers
asked Mar 28 at 2:40
ElliottElliott
947
asked Mar 28 at 2:40
ElliottElliott
947
asked Mar 28 at 2:40
ElliottElliott
947
asked Mar 28 at 2:40
ElliottElliott
947
asked Mar 28 at 2:40
ElliottElliott
947
947
add a comment |
add a comment |
1 Answer
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You're right. The linked answer is in error; it's using the reverse Fibonacci recurrence $f_k+1+f_k=f_k-1$ consistently, and not noticing that this isn't quite the same as the original.
answered Mar 28 at 3:15
jmerryjmerry
16.9k11633
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$begingroup$
You're right. The linked answer is in error; it's using the reverse Fibonacci recurrence $f_k+1+f_k=f_k-1$ consistently, and not noticing that this isn't quite the same as the original.
answered Mar 28 at 3:15
jmerryjmerry
16.9k11633
$endgroup$
add a comment |
$begingroup$
You're right. The linked answer is in error; it's using the reverse Fibonacci recurrence $f_k+1+f_k=f_k-1$ consistently, and not noticing that this isn't quite the same as the original.
answered Mar 28 at 3:15
jmerryjmerry
16.9k11633
$endgroup$
add a comment |
$begingroup$
You're right. The linked answer is in error; it's using the reverse Fibonacci recurrence $f_k+1+f_k=f_k-1$ consistently, and not noticing that this isn't quite the same as the original.
answered Mar 28 at 3:15
jmerryjmerry
16.9k11633
$endgroup$
You're right. The linked answer is in error; it's using the reverse Fibonacci recurrence $f_k+1+f_k=f_k-1$ consistently, and not noticing that this isn't quite the same as the original.
answered Mar 28 at 3:15
jmerryjmerry
16.9k11633
answered Mar 28 at 3:15
jmerryjmerry
16.9k11633
answered Mar 28 at 3:15
jmerryjmerry
16.9k11633
answered Mar 28 at 3:15
jmerryjmerry
16.9k11633
16.9k11633
add a comment |
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