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2. Can you identify how Fibonacci numbers are used in Pascal's Triangle?
1
11
121
1331
14641
1 5 10 10 5 1
16 15 20 15 6 1
1 7 21 35 35 21 71
​

Sagot :

Nobodyisperfectsidn

Fibonacci numbers appear in Pascal's Triangle along the diagonals. Here's how you can identify them:

1. Pascal's Triangle is constructed such that each number is the sum of the two numbers directly above it.

2. To find Fibonacci numbers, start with the top row and then sum numbers along the shallow diagonals.

Here's a step-by-step guide:

1. Label the rows of Pascal's Triangle starting with 0 for the topmost row.

2. Look at the diagonals from left to right.

For example, consider the following part of Pascal's Triangle:

```

1 ← Row 0

1 1 ← Row 1

1 2 1 ← Row 2

1 3 3 1 ← Row 3

1 4 6 4 1 ← Row 4

1 5 10 10 5 1 ← Row 5

```

If you sum the numbers along the shallow diagonals, you'll find the Fibonacci sequence:

- The first diagonal is just 1.

- The second diagonal is 1, 1 (sum: 1).

- The third diagonal is 1, 2 (sum: 1 + 1 = 2).

- The fourth diagonal is 1, 3, 1 (sum: 1 + 2 = 3).

- The fifth diagonal is 1, 4, 3 (sum: 1 + 3 = 4, 4 + 1 = 5).

Summing these shallow diagonals gives you the Fibonacci numbers:

```

1 ← 1

1 1 ← 1

1 2 1 ← 2

1 3 3 1 ← 3

1 4 6 4 1 ← 5

1 5 10 10 5 1 ← 8

```

So, the Fibonacci numbers are formed by summing up these diagonals.

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