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8.5D The Nth Term of a Quadratic Sequence: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculator allowed · about 50 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Write down the second difference for the sequence 4, 9, 16, 25, 36.

(Total for Question 1 is 1 mark)

The quadratic sequence begins 3, 8, 15, 24, 35. Find an expression for the nth term in the form an² + bn + c.

(Total for Question 2 is 3 marks)

Work out an expression for the nth term of the sequence 5, 12, 21, 32, 45.

(Total for Question 3 is 2 marks)

Write down the first four terms of the sequence defined by T(n) = 2n² - n + 3.

(Total for Question 4 is 2 marks)

The sequence 5, 11, 19, 29, 41,... is generated by a quadratic rule. Show that the nth term is n² + 3n + 1.

(Total for Question 5 is 3 marks)

The nth term of a sequence is n² + 4n + 3. Find the value of n for which the term is 63.

(Total for Question 6 is 3 marks)

A quadratic sequence has constant second difference 8. Its 3rd term is 24 and its 5th term is 60. Find an expression for the nth term.

(Total for Question 7 is 3 marks)

The sequence begins 10, 24, 44, 70, 102. (a) Find an expression for the nth term. (b) Hence find the value of n for which the term is 70.

(a)Find an expression for the nth term.(2)

(b)Hence find the value of n for which the term is 70.(2)

(Total for Question 8 is 4 marks)

Find all integer values of n for which n² - 3n + 2 = 0.

(Total for Question 9 is 2 marks)

The nth term of a sequence is 4n² - 7n + 2. Find the smallest positive integer n for which the term is greater than 200.

(Total for Question 10 is 3 marks)

Find an expression for the nth term of the sequence 2, 9, 20, 35.

(Total for Question 11 is 2 marks)

Find the nth term of the sequence 8, 21, 40, 65.

(Total for Question 12 is 2 marks)

Work out an expression for the nth term of the sequence 1, 6, 15, 28.

(Total for Question 13 is 2 marks)

The 2nd term of a quadratic sequence is 30, the 4th term is 78 and the 6th term is 150. (a) Find an expression for the nth term. (b) Find the sum of the first 6 terms of the sequence.

(a)Find an expression for the nth term.(3)

(b)Find the sum of the first 6 terms.(2)

(Total for Question 14 is 5 marks)

Mark scheme · 8.5D The Nth Term of a Quadratic Sequence: Fluency and Exam Drill

Question 1

B1 second difference = 2 cao
Answer: 2

Question 2

M1 second difference = 2 so a = 1 (2a = 2) oe
M1 use n = 1 and n = 2 to form linear equations and solve for b and c (eg 1 + b + c = 3 and 4 + 2b + c = 8) oe
A1 nth term = n² + 2n cao
Answer: n² + 2n

Question 3

M1 second difference = 2 so a = 1 (2a = 2) oe
A1 nth term = n² + 4n cao
Answer: n² + 4n

Question 4

M1 substitute n = 1, 2, 3, 4 into 2n² - n + 3 oe
A1 4, 9, 18, 31 cao
Answer: 4, 9, 18, 31

Question 5

M1 second difference = 2 so a = 1 (2a = 2) oe
M1 use n = 1 and n = 2 to find b and c (1 + b + c = 5 and 4 + 2b + c = 11) oe
A1 nth term = n² + 3n + 1 cao cso
Answer: n² + 3n + 1

Question 6

M1 form equation n² + 4n + 3 = 63 oe
M1 rearrange and factorise to (n + 10)(n - 6) = 0 or equivalent method
A1 n = 6 cao (positive integer solution)
Answer: 6

Question 7

M1 second difference = 8 so a = 4 (2a = 8) oe
M1 use T3 and T5 to form two linear equations and solve for b and c (eg 36 + 3b + c = 24 and 100 + 5b + c = 60) oe
A1 nth term = 4n² - 14n + 30 cao
Answer: 4n² - 14n + 30

Question 8

(a) M1 second differences 14, 20, 26 so second difference = 6 and a = 3 (2a = 6) oe
(a) A1 nth term = 3n² + 5n + 2 cao
(a) Answer: 3n² + 5n + 2
(b) M1 set 3n² + 5n + 2 = 70 and rearrange to 3n² + 5n - 68 = 0 oe
(b) A1 n = 4 cao
(b) Answer: 4

Question 9

M1 factorise to (n - 1)(n - 2) = 0 or equivalent
A1 n = 1 or n = 2 cao
Answer: n = 1 or n = 2

Question 10

M1 set inequality 4n² - 7n + 2 > 200 and rearrange to 4n² - 7n - 198 > 0 oe
M1 find positive root of 4n² - 7n - 198 = 0 approximately (n approx 7.97) oe
A1 smallest integer n = 8 cao
Answer: 8

Question 11

M1 second differences 7, 11, 15 so second difference = 4 and a = 2 (2a = 4) oe
A1 nth term = 2n² + n - 1 cao
Answer: 2n² + n - 1

Question 12

M1 second differences 13, 19, 25 so second difference = 6 and a = 3 (2a = 6) oe
A1 nth term = 3n² + 4n + 1 cao
Answer: 3n² + 4n + 1

Question 13

M1 second differences 5, 9, 13 so second difference = 4 and a = 2 (2a = 4) oe
A1 nth term = 2n² - n cao
Answer: 2n² - n

Question 14

(a) M1 form three equations 4a + 2b + c = 30, 16a + 4b + c = 78, 36a + 6b + c = 150 and use second difference or elimination to find a
(a) M1 solve to find b after a is known (eg a = 3 then b = 6) oe
(a) A1 nth term = 3n² + 6n + 6 cao
(a) Answer: 3n² + 6n + 6
(b) M1 use sum S = sum_{n=1..6}(3n² + 6n + 6) = 3 sum n² + 6 sum n + 6*6 oe
(b) A1 S = 435 cao
(b) Answer: 435

The Nth Term of a Quadratic Sequence: Fluency and Exam Drill - Worksheets, Questions and Revision