Question 1: The Difference Method involves calculating the successive differences of a sequence to identify if it is quadratic. If the second differences are constant, the sequence is quadratic. Use this method to find the nth term of the sequence.

Answer: Use second differences to determine the quadratic formula.

Question 1: Given the sequence 3, 8, 15, 24, find the second differences.

Answer: Constant; 2

Question 2: Determine whether the sequence 5, 12, 21, 32, is quadratic using the difference method.

Answer: Yes

Question 3: Find the second difference for the sequence 2, 6, 12, 20, 30.

Answer: Constant; 4

Question 4: Calculate the third difference for the sequence 1, 4, 9, 16, 25.

Answer: Zero

Question 1: A ball is dropped from a height, and its height after n seconds follows a quadratic sequence. The heights after 1, 2, 3, and 4 seconds are 45m, 80m, 117m, and 156m respectively. Use the difference method to find the nth term for the height in meters.

Answer: h(n) = 15n^2 + 30n

Question 2: A company records the profit over four years as 200, 350, 500, 650 dollars. Determine if the profit sequence is quadratic and find the formula if it is.

Answer: Yes; profit = 150n + 50

Question 1: A gardener plants a new type of tree, and the height of the tree in cm after n years follows the sequence 30, 70, 130, 210. Use the difference method to model the height of the tree after n years and predict the height after 5 years.

Answer: Height = 20n^2 + 10n, so after 5 years, height = 20(5)^2 + 10(5) = 20*25 + 50 = 550 cm

Question 2: A car's acceleration is modeled such that its distance from a starting point after n seconds is given by the sequence 0, 5, 18, 37, 62. Use the difference method to find the general formula for the distance traveled.

Answer: Distance = 2n^2 + 3n

Question 1: Given the sequence 4, 14, 28, 46, determine if it is quadratic using the difference method. If so, find the nth term.

Answer: Yes; nth term = 4n^2 - 2n

Question 2: A basketball's scoring pattern over games is 10, 20, 35, 50, 65. Use the difference method to determine if this is quadratic and find the formula.

Answer: Yes; score = 7.5n^2 + 2.5n

Question 1: A sequence shows constant first differences but is claimed to be quadratic. Is this correct? Explain.

Answer: No; constant first differences indicate linear, not quadratic.

Question 2: A student finds the second differences as 3, 3, 3 but makes an arithmetic mistake and writes the sequence as quadratic. Identify and correct the mistake.

Answer: They correctly identified constant second differences; mistake could be in calculating the sequence's nth term or missing the constant second difference.