Question
Given that m is an integer,
(a) write down expressions for the next two odd numbers after 2n - 1,
b) © find, in its simplest form, the expression for the sum of these three odd numbers,
(©) explain why the sum is a multiple of 3,
(c) find, in its simplest form, an expression for the sum of the squares of these three odd
numbers.
(a) write down expressions for the next two odd numbers after 2n - 1,
b) © find, in its simplest form, the expression for the sum of these three odd numbers,
(©) explain why the sum is a multiple of 3,
(c) find, in its simplest form, an expression for the sum of the squares of these three odd
numbers.
Answers
mathhelper
a) they would have to be 2 units apart, so:
2n-1, 2n+1, 2n+3
b) sum of those = 2n-1 + 2n+1 + 2n+3 = 6n +3
we can factor out 3 to get 3(2n+1) which is a multiple of 3, since 3 is a factor
c) sum of squares = (2n-1)^2 + (2n+1)^2 + (2n+3)^2
= 4n^2 - 4n + 1 + 4n^2 + 4n + 1 + 4n^2 + 12n + 9
= 12n^2 + 12n + 11
2n-1, 2n+1, 2n+3
b) sum of those = 2n-1 + 2n+1 + 2n+3 = 6n +3
we can factor out 3 to get 3(2n+1) which is a multiple of 3, since 3 is a factor
c) sum of squares = (2n-1)^2 + (2n+1)^2 + (2n+3)^2
= 4n^2 - 4n + 1 + 4n^2 + 4n + 1 + 4n^2 + 12n + 9
= 12n^2 + 12n + 11