Answer:
The expression of the pattern is n² + 2
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