from principle of mathametical induction prove that for all positive integral values of n:(n^2-1) is devisible by 24 , where n is odd positive integer
Answers
How does one mathematically prove that n2−1 is divisible by 8, so long as n is an odd integer? Is this sufficient as a mathematical proof?
We can prove it by taking values of n. This will also give the answer. But, we can prove it in general form for all possible values of n, provided n is a natural number. Here is the solution in which, n is taken in general form, not any specific value. This is the safest and authentic way to prove the question. This is true for all the values of n (Natural number). This method is very simple and can be understood by anyone. Let’s see the method.
A simple solution:
Here n is odd. So, we can write it in the form of n = 2k+1.
Now, let’s put it in the original equation:
n2–1
= (2k+1)2–1
= (4k2+4k+1)–1
= 4k2+4k
= 4k(k+1)
Now, there are 2 possibilities:
k is odd
k is even
In both the cases, it is divisible by 8, which is clearly visible in the image-
