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 (2n−1)2+(2n+1)2=8n2+2, where n∈Z.
Hence, or otherwise, prove that the sum of the squares of any two consecutive odd integers is even.
Markscheme
attempting to expand the LHS (M1)
LHS =(4n2−4n+1)+(4n2+4n+1) A1
=8n2+2 (= RHS) AG
[2 marks]
METHOD 1
recognition that 2n−1 and 2n+1 represent two consecutive odd integers (for n∈Z) 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 n∈Z) 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]