Question

Carnegie Learning, Inc

Answer 1

Answer:

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