1 Introduction
2 Main results
3 Examples
n
|
Theorem
2.1
|
Theorem
1.3
|
\(\frac{\mathbf{Third\ column}}{\mathbf{Second\ column}}\)
|
---|---|---|---|
2 | 1 | 1 |
\(\frac{1}{1}=1\)
|
3 | 2 | 2 |
\(\frac{2}{2}=1\)
|
4 |
\(3\sqrt{2}\)
| 6 |
\(\frac{6}{3\sqrt{2}}=\sqrt{2}\)
|
5 |
\(\sqrt{60}\)
| 15 |
\(\frac{15}{\sqrt{60}}\approx1.936\)
|
6 |
\(\sqrt{200}\)
| 40 |
\(\frac{40}{\sqrt{200}}=2\sqrt{2}\)
|
n
|
\(\sqrt{(n-1)F_{n}F_{n-1}}\)
|
\(F_{n}F_{n-1}\)
|
\(\frac {F_{n}F_{n-1}}{\sqrt{(n-1)F_{n}F_{n-1}}}=\sqrt{\frac {F_{n}F_{n-1}}{n-1}}\)
|
n
|
Theorem
2.4
|
Theorem
1.4
|
\(\frac{\mathbf{Third\ column}}{\mathbf{Second\ column}}\)
|
---|---|---|---|
1 | 2 | 2 |
\(\frac{2}{2}=1\)
|
2 |
\(\sqrt{10}\)
|
\(\sqrt{10}\)
|
\(\frac{\sqrt{10}}{\sqrt{10}}=1\)
|
3 |
\(\sqrt{42}\)
|
\(\sqrt{154}\)
|
\(\frac{\sqrt{154}}{\sqrt {42}}\approx 1.915\)
|
4 |
\(\sqrt{120}\)
|
\(\sqrt{810}\)
|
\(\frac{\sqrt{810}}{\sqrt {120}}\approx 2.598\)
|
5 |
\(\sqrt{395}\)
|
\(\sqrt{6{,}004}\)
|
\(\frac{\sqrt{6{,}004}}{\sqrt {395}}\approx3.899\)
|
6 |
\(\sqrt{1{,}200}\)
|
\(\sqrt{39{,}400}\)
|
\(\frac{\sqrt{39{,}400}}{\sqrt {1{,}200}}\approx5.730\)
|
n
|
\(\sqrt{n^{-1}(5F_{n}F_{n-1}+1)}\)
n odd,
\(\sqrt{n^{-1}(5F_{n}F_{n-1}-3)}\)
n even |
n
|
Theorem
2.8
|
Theorem
1.5
|
\(\frac{\mathbf{Third\ column}}{\mathbf{Second\ column}}\)
|
---|---|---|---|
2 | 2 | 2 |
\(\frac{2}{2}=1\)
|
3 | 4 | 4 |
\(\frac{4}{4}=1\)
|
4 |
\(6\sqrt{2}\)
| 12 |
\(\frac{12}{6\sqrt{2}}=\sqrt{2}\)
|
5 |
\(4\sqrt{15}\)
|
\(\sqrt{30}\)
|
\(\frac{30}{4\sqrt{15}}\approx 1.936\)
|
6 |
\(20\sqrt{2}\)
| 80 |
\(\frac{80}{20\sqrt{2}}=2\sqrt{2}\)
|
n
|
\(\sqrt{(n-1)|r|^{2}F_{n}F_{n-1}}\)
|
\(|r|F_{n}F_{n-1}\)
|
\(\sqrt {(n-1)^{-1}F_{n}F_{n-1}}\)
|
n
|
Theorem
2.12
|
Theorem
1.6
|
\(\frac{\mathbf{Third\ column}}{\mathbf{Second\ column}}\)
|
---|---|---|---|
1 | 2 | 2 |
\(\frac{2}{2}=1\)
|
2 | 5 | 4 |
\(\frac{4}{5}=\frac{4}{5}\)
|
3 |
\(3\sqrt{14}\)
|
\(2\sqrt{231}\)
|
\(\frac{2\sqrt{231}}{3\sqrt {14}}\approx2.708\)
|
4 |
\(\sqrt{390}\)
| 54 |
\(\frac{54}{\sqrt{390}}\approx2.734\)
|
5 |
\(\sqrt{1{,}343}\)
| 152 |
\(\frac{152}{\sqrt{1{,}343}}\approx4.418\)
|
6 |
\(10\sqrt{42}\)
| 394 |
\(\frac{394}{10\sqrt{342}}\approx6.080\)
|