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8.3D Proof: Fluency and Exam Drill

EDEXCEL Edexcel 1MA1 · Calculator allowed · about 60 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Prove that n² - n is even for any integer n.

(Total for Question 1 is 2 marks)

Prove that n² + n is even for any integer n.

(Total for Question 2 is 2 marks)

Show that the product of three consecutive integers n(n + 1)(n + 2) is divisible by 3 for any integer n.

(Total for Question 3 is 2 marks)

Prove that n³ - n is divisible by 6 for any integer n.

(Total for Question 4 is 3 marks)

Prove that n and n² have the same parity. That is, show n is even if and only if n² is even, and n is odd if and only if n² is odd.

(Total for Question 5 is 3 marks)

Show that the sum of three consecutive integers n + (n + 1) + (n + 2) is divisible by 3 for any integer n.

(Total for Question 6 is 3 marks)

Prove that n³ + 5n is divisible by 6 for any integer n.

(Total for Question 7 is 4 marks)

Let n be an odd integer. Prove that n² - 1 is divisible by 8.

(Total for Question 8 is 4 marks)

Show that the difference of consecutive square numbers (n + 1)² - n² is odd for any integer n.

(Total for Question 9 is 2 marks)

Stretch question. Prove that n⁵ - n is divisible by 30 for any integer n.

(Total for Question 10 is 5 marks)

Mark scheme · 8.3D Proof: Fluency and Exam Drill

Question 1

M1 factorise to n(n - 1) or show product of two consecutive integers oe
A1 conclude n(n - 1) is even, so n² - n is even cao
Answer: n(n - 1) is even, so n² - n is even (for all integers n).

Question 2

M1 factorise to n(n + 1) or show product of two consecutive integers oe
A1 conclude n(n + 1) is even, so n² + n is even cao
Answer: n(n + 1) is even, so n² + n is even (for all integers n).

Question 3

M1 note that among n, n + 1, n + 2 one is a multiple of 3 oe
A1 conclude n(n + 1)(n + 2) is divisible by 3 cao
Answer: n(n + 1)(n + 2) is divisible by 3.

Question 4

M1 factorise to n(n - 1)(n + 1) oe
M1 state that among three consecutive integers one is even and one is a multiple of 3 (so product divisible by 2 and by 3)
A1 conclude n³ - n is divisible by 6 cao
Answer: n³ - n = n(n - 1)(n + 1) is divisible by 6.

Question 5

M1 show if n = 2k then n² = 4k² which is even (even case)
M1 show if n = 2k + 1 then n² = 4k² + 4k + 1 which is odd (odd case)
A1 conclude n and n² have the same parity cao
Answer: If n = 2k then n² is even; if n = 2k + 1 then n² is odd. So n and n² have the same parity.

Question 6

M1 sum the three terms to get 3n + 3
M1 factorise to 3(n + 1)
A1 conclude the sum is divisible by 3 cao
Answer: n + (n + 1) + (n + 2) = 3(n + 1), so divisible by 3.

Question 7

M1 rewrite n³ + 5n as (n³ - n) + 6n
M1 show n³ - n = n(n - 1)(n + 1), which is divisible by 6 (product of three consecutive integers)
M1 state 6n is divisible by 6
A1 conclude n³ + 5n is divisible by 6 cao
Answer: n³ + 5n = (n³ - n) + 6n, both parts divisible by 6, so n³ + 5n divisible by 6.

Question 8

M1 substitute n = 2k + 1 and expand: n² - 1 = (2k + 1)² - 1
M1 simplify to 4k(k + 1)
M1 state k(k + 1) is even (product of consecutive integers) so has factor 2
A1 conclude n² - 1 = 4k(k + 1) is divisible by 8 cao
Answer: If n = 2k + 1 then n² - 1 = 4k(k + 1). Since k(k + 1) is even, n² - 1 is divisible by 8.

Question 9

M1 calculate (n + 1)² - n² = 2n + 1
A1 state 2n + 1 is odd, so the difference is odd cao
Answer: (n + 1)² - n² = 2n + 1, which is odd for any integer n.

Question 10

M1 factorise n⁵ - n = n(n⁴ - 1) = n(n² - 1)(n² + 1) = n(n - 1)(n + 1)(n² + 1)
M1 state among n - 1, n, n + 1 one is even, so there is a factor 2
M1 state among n - 1, n, n + 1 one is multiple of 3, so there is a factor 3
M1 show divisibility by 5 by checking n mod 5 = 0, 1, 2, 3, 4 (or use Fermat): in each case n⁵ - n is a multiple of 5
A1 conclude n⁵ - n is divisible by 2, 3 and 5 and hence by 30 cao
Answer: n⁵ - n = n(n - 1)(n + 1)(n² + 1) is divisible by 2 and 3 from the three consecutive factors, and by 5 by case or Fermat argument; therefore divisible by 30.

Proof: Fluency and Exam Drill - Worksheets, Questions and Revision