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.
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
a) You are correct in saying that the fourth odd number can be expressed as 7 = 4^2 - 3^2. You can calculate this by squaring the two consecutive integers, 4 and 3, and subtracting the result of squaring the smaller number from the larger number.
b) However, you made a mistake in expressing the number 19. 19 cannot be written as a difference of squares. To express it in this form, we need two integers whose squares have a difference of 19. Since it is not possible to find such two squares, we cannot express 19 in this specific form.
c) To write a formula for the nth odd number in this form, let's analyze the pattern in the given examples.
The first odd number is 1 = 1^2 - 0^2.
The second odd number is 3 = 2^2 - 1^2.
The third odd number is 5 = 3^2 - 2^2.
From this pattern, we can observe that the nth odd number can be expressed as n^2 - (n-1)^2.
Simplifying this expression:
n^2 - (n-1)^2 = n^2 - (n^2 - 2n + 1)
= n^2 - n^2 + 2n - 1
= 2n - 1
Therefore, the formula for the nth odd number is 2n - 1.
d) To prove that the product of two consecutive odd numbers is always odd, let's consider two consecutive odd numbers as (2n-1) and (2n+1), where n is an integer.
The product of these two consecutive odd numbers would be:
(2n-1)(2n+1) = 4n^2 - 1
We can observe that 4n^2 is an even number since it is divisible by 2. Adding or subtracting 1 from an even number always results in an odd number.
Therefore, (4n^2 - 1) is always an odd number.
Hence, the product of two consecutive odd numbers is always odd.