### 7.4 Fibonacci & Other Patterns

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• 15. Triangular Numbers

The sequence of triangular numbers is

1, 3, 6, 10, 15, 21, 28, 36, 45, ....

Describe the pattern. (See Example 5 and Example 6.)

• Worked-Out Solution

The nth triangular number is the sum of the first n whole numbers.

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``` __   __     ___      _____      ___      _____
\ \\/ //   / _ \\   |__  //    / _ \\   |__  //
\   //   / //\ \\    / //    / //\ \\    / //
/ . \\  |  ___  ||  / //__  |  ___  ||  / //__
/_//\_\\ |_||  |_|| /_____|| |_||  |_|| /_____||
`-`  --` `-`   `-`  `-----`  `-`   `-`  `-----`

```
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• 16. Triangular Numbers

The sequence of triangular numbers is

1, 3, 6, 10, 15, 21, 28, 36, 45, ....

Describe how the sequence of triangular numbers is related to the following diagram. (See Example 5 and Example 6.)

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``` _____       ___     __   __   _    _    __   __
|  __ \\    / _ \\   \ \\/ // | || | ||  \ \\/ //
| |  \ ||  | / \ ||   \   //  | || | ||   \   //
| |__/ ||  | \_/ ||   / . \\  | \\_/ ||   / . \\
|_____//    \___//   /_//\_\\  \____//   /_//\_\\
-----`     `---`    `-`  --`   `---`    `-`  --`

```
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• 17. Triangular Numbers

The sequence of triangular numbers is

1, 3, 6, 10, 15, 21, 28, 36, 45, ....

Is 115 a triangular number? Explain your reasoning. (See Example 5 and Example 6.)

• Worked-Out Solution

The nth triangular number is the sum of the first n whole numbers.

By continuing the pattern, you can see that 115 is not a triangular number.

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```  _____      ___      ______    _____    __   _
/ ____||   / _ \\   /_____//  |  ___|| | || | ||
/ //---`'  | / \ ||  `____ `   | ||__   | '--' ||
\ \\___    | \_/ ||  /___//    | ||__   | .--. ||
\_____||   \___//   `__ `     |_____|| |_|| |_||
`----`    `---`    /_//      `-----`  `-`  `-`
`-`
```
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• 18. Lucas Sequence

The Lucas sequence is named in honor of mathematician François Édouard Anatole Lucas.

The Lucas sequence is

1, 3, 4, 7, 11, 18, 29, 47, 76,....

Describe the pattern. (See Example 5 and Example 6.)

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```  _  __   __   __     _____    _____     _____
| |/ //  \ \\/ //   / ___//  |  ___||  / ____||
| ' //    \ ` //    \___ \\  | ||__   / //---`'
| . \\     | ||     /    //  | ||__   \ \\___
|_|\_\\    |_||    /____//   |_____||  \_____||
`-` --`    `-`'   `-----`    `-----`    `----`

```
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• 19. Lucas Sequence

The Lucas sequence is named in honor of mathematician François Édouard Anatole Lucas.

The Lucas sequence is

1, 3, 4, 7, 11, 18, 29, 47, 76, ....

Describe how the Lucas sequence is related to the cactus. (See Example 5 and Example 6.)

• Worked-Out Solution

When looking down on the cactus, there are two sets of spirals. One is clockwise and the other is counterclockwise. The one set has 18 spirals and the other set has 11 spirals.

The numbers 11 and 18 are the 5th and 6th numbers of the Lucas sequence.

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```  ______     ___     _    _      ___
/_____//   / _ \\  | |  | ||   / _ \\      ___
`____ `   | / \ || | |/\| ||  / //\ \\    /   ||
/___//    | \_/ || |  /\  || |  ___  ||  | [] ||
`__ `      \___//  |_// \_|| |_||  |_||   \__ ||
/_//       `---`   `-`   `-` `-`   `-`     -|_||
`-`                                         `-`
```
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• 20. Lucas Sequence

Use the spreadsheet to find the ratio of successive Lucas numbers. What number do you approach as the numbers get larger? (See Example 5 and Example 6.)

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```__    __     ___      _  _    __   __   __   _
\ \\ / //   / _ \\   | \| ||  \ \\/ // | || | ||
\ \/ //   | / \ ||  |  ' ||   \ ` //  | '--' ||
\  //    | \_/ ||  | .  ||    | ||   | .--. ||
\//      \___//   |_|\_||    |_||   |_|| |_||
`       `---`    `-` -`     `-`'   `-`  `-`

```