Question 1: Given the sequence 3, 8, 15, 24, find the second differences and verify if it's quadratic.

Answer: Second differences are constant at 2; sequence is quadratic

Question 2: Calculate the first and second differences of the sequence 2, 7, 14, 23, 34.

Answer: First differences: 5, 7, 9, 11; Second differences: 2, 2, 2

Question 3: Construct the next term in the sequence 5, 11, 19, 29 using difference method.

Answer: Next term is 41

Question 1: A sequence of numbers has a constant second difference of 4. If the first term is 2, find the sequence's fifth term using the difference method.

Answer: 33

Question 2: Explain why a sequence with constant first differences is not quadratic, but one with constant second differences is.

Answer: Constant first differences indicate a linear sequence; constant second differences indicate a quadratic sequence because second differences are related to the quadratic term's coefficient.

Question 3: Given the sequence 1, 4, 9, 16, identify the mistake if a student claims the sequence is arithmetic using difference method.

Answer: The sequence has increasing differences (3, 5, 7), so it is not arithmetic; mistake is assuming constant differences.

Question 1: A ball is dropped from a height where its height after each bounce forms a quadratic sequence. The heights after successive bounces are 16, 9, 4, 1. Use the difference method to predict the height after the fifth bounce.

Answer: 0

Question 2: A car accelerates so that its distance covered over equal time intervals forms a quadratic sequence. The distances are 3 km, 8 km, 15 km, 24 km. Using the difference method, determine the distance after the 5th interval.

Answer: 35 km

Question 1: A quadratic sequence has a second difference of 6. The first term is 2, and the second term is 7. Find the explicit formula for the sequence using the difference method.

Answer: a_n = n^2 + 1

Question 2: Given a sequence with constant second difference of -2 and first term 10, find the third term.

Answer: 4

Question 1: A student claims that all sequences with constant differences are quadratic. Identify the mistake and give an example of a sequence with constant differences that is not quadratic.

Answer: Error: Constant differences only indicate linear sequences, not quadratic. Example: 1, 2, 3, 4 (arithmetic, not quadratic)

Question 2: A sequence's second differences are calculated as 3, 3, 3, but a student incorrectly concludes the sequence is cubic. What is the mistake? Correct the misconception.

Answer: Mistake: Constant second differences mean quadratic, not cubic. Cubic sequences have constant third differences.