Question

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.

Answer 1

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)(M-1) is the product of two odd numbers
= NM+M+N-1
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