Question 1: The nth term of a quadratic sequence can often be written in the form an^2 + bn + c. To find the nth term, students need to analyze the pattern of the sequence and determine the coefficients a, b, and c.

Answer:

Question 1: Calculate the nth term of the sequence: 3, 8, 15, 24, ...

Answer: n^2 + 2n + 0

Question 2: Find the nth term of the sequence: 2, 6, 12, 20, ...

Answer: n^2 + n + 0

Question 3: Identify the mistake in the following attempt to find the nth term of sequence 5, 14, 27, 44,...: ‘The differences are 9, 13, 17, so the sequence is quadratic with nth term n^2 + 4n + 0’.

Answer: Incorrect: The difference of differences is not constant; the sequence is not quadratic.

Question 4: Given the sequence 1, 6, 15, 28, ... , derive the nth term.

Answer: n^2 + 2n - 2

Question 5: The sequence 4, 9, 16, 25, ... is quadratic. Find its nth term.

Answer: n^2 + 3

Question 6: A student states, ‘The nth term of a quadratic sequence always has the form 2n^2 + 3n + 1’. Is this always true? Justify your answer.

Answer: No, the form depends on the specific sequence; the coefficients vary.

Question 7: Construct a quadratic sequence with the nth term n^2 + 4.

Answer: Sequence: 5, 10, 17, 26, ...

Question 8: A sequence has the nth term n^2 + 2n + 1. List the first four terms and verify the formula.

Answer: Terms: 4, 9, 16, 25. Confirmed by plugging n=1,2,3,4.

Question 9: Explain why the sequence 2, 6, 12, 20, ... is quadratic and find its nth term.

Answer: Differences of differences are constant; nth term is n^2 + n.

Question 10: Identify and correct the mistake: A student finds the nth term as 2n^2 + 3n + 2 for the sequence 1, 4, 9, 16.

Answer: Correct nth term should be n^2; student’s formula is incorrect.

Question 11: Given the sequence 7, 19, 37, 61, ... , find the nth term and explain your reasoning.

Answer: Sequence differences suggest quadratic; nth term is 2n^2 + 3n + 2.