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Date May Specimen paper Marks available 2 Reference code SPM.1.SL.TZ0.3
Level Standard Level Paper Paper 1 (without calculator) Time zone Time zone 0
Command term Show that Question number 3 Adapted from N/A

Question

Show that (2n1)2+(2n+1)2=8n2+2, where nZ.

[2]
a.

Hence, or otherwise, prove that the sum of the squares of any two consecutive odd integers is even.

[3]
b.

Markscheme

attempting to expand the LHS   (M1)

LHS =(4n24n+1)+(4n2+4n+1)     A1

=8n2+2 (= RHS)    AG

[2 marks]

a.

METHOD 1

recognition that 2n1 and 2n+1 represent two consecutive odd integers (for nZ)      R1

8n2+2=2(4n2+1)     A1

valid reason eg divisible by 2 (2 is a factor)       R1

so the sum of the squares of any two consecutive odd integers is even        AG

 

METHOD 2

recognition, eg that n and n+2 represent two consecutive odd integers (for nZ)       R1

n2+(n+2)2=2(n2+2n+2)     A1

valid reason eg divisible by 2 (2 is a factor)       R1

so the sum of the squares of any two consecutive odd integers is even        AG

[3 marks]

b.

Examiners report

[N/A]
a.
[N/A]
b.

Syllabus sections

Topic 1—Number and algebra » SL 1.6—Simple proof
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Topic 1—Number and algebra

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