Posted by Stephanie on Thursday, November 20, 2008 at 1:16pm.
The first odd number can be expressed as 1 = 1squared  0squared.
The second odd number can be expressed as 3 = 2squared  1squared.
The third odd number can be expressed as 5 = 3squared  2squeared.
a) Express the fourth odd number in this form. (Am I right in saying 7 = 4squared  3squared)
b) Express the number 19 in this form. (Am I right in saying 19 = 10squared  9squared)
c) Write down a formula for the Nth odd number and simplify this expression.
d) PROVE that the product of two consecutive odd numbers is always odd.

Maths  bobpursley, Thursday, November 20, 2008 at 1:23pm
yes, on a,b.
c) N=((INT N/2)+1)^2(INT(N/2))^2
d)Let N be and even number, and M an even number.
(N+1)(M1) is the product of two odd numbers
= NM+M+N1
BUt NM is even (the product of two evens), M is even, N is even, so NM+M+N is even. If one subtracted, then the product of two odd numbers is odd.
If you cannot use the arguement NM is even, then consider
NM= M+M+M+....M where the M is added N times. The sum of even number is even.
QED
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