Carnegie Learning, Inc

3. Charlie watches as a pattern develops. He starts with 3 guppy fish in his fish tank. The next

month he finds 6 guppy fish in his tank. The following month, he finds 11 guppy fish in his fish tank.

Write an expression to model the pattern of guppy fish in Charlie's fish tank.

The expression of the pattern is n² + 2

Step-by-step explanation:

* Lets explain how to solve the problem

- Charlie starts with 3 guppy fish in his tank

- After one month he finds 6 guppy fish in his tank

- The following month, he finds 11 guppy fish in his tank

∵ The increasing in the next month is ⇒ 6 - 3 = 3

∵ The increasing in the following month is ⇒ 11 - 6 = 5

- The number is increasing by the consecutive odd numbers

∴ The nth rule is quadratic ⇒ an² + bn + c, where

a + b + c = 1st term

3a - b = the first difference

2a = the constant difference between each two difference

∵ The first term is 3

∴ a + b + c = 3

∵ The first difference is 6 - 3 = 3

∴ 3a - b = 3

∵ 6 - 3 = 3 and 11 - 6 = 5

∵ 5 - 3 = 2

∴ The constant difference is 2

∴ 2a = 2 ⇒ divide both sides by 2

∴ a = 1

∵ 3a - b = 3

∴ 3 (1) - b = 3

∴ 3 - b = 3 ⇒ subtract 3 from both sides

∴ b = 0

∵ a + b + c = 3

∴ 1 + 0 + c = 3

∴ 1 + c = 3 ⇒ subtract 1 from both sides

∴ c = 2

∵ The nth rule is ⇒ an² + bn + c, where n is the position of the number

∴ The nth rule = (1) n² + (0) n + 2

∴ The nth rule = n² + 2

∴ The expression of the pattern is n² + 2