Algebra 9
2 marks · HardProve that for any integer , the sum is always an odd number.
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The sum is always odd for any integer n.
(a) 2n^2 + 2n + 1 is always odd for integer n. · must show Expand the sum, State the result is always odd
Must show: Expand and combine terms, Explain parity of result
Common mistakes: Assuming and have same parity; Expanding incorrectly
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Marking points
M1 · 1 mark · Expands to .
A1 · 1 mark · Clearly states and justifies why the result is always odd for any integer .
Worked solution
Expand the sum:
No matter what integer value has, is always even (since both terms have a factor of 2). Adding 1 makes it odd. Thus, is always odd for any integer .