Question

n^2>=n prove that when n>=1​

Answer 1

Solution :

We can prove it using the method of Induction.

Step 1.

When n = 1,

1² = 1

i.e., the given statement holds for n = 1

Step 2.

When n = 2,

2² = 4 > 2

When n = 3,

3² = 9 > 3

and so on

i.e., the given statement holds for n = 2, 3, ...

Step 3.

Let, the given statement holds for n = k (> 0)

Then, k² > k

Now, (k + 1)² - (k + 1)

= k² + 2k + 1 - k - 1

= k² + k + 1

> 2k + 1 , as k² > k

> 0, as k > 0

i.e., (k + 1)² - (k + 1) > 0

i.e., (k + 1)² > k + 1

Thus the statement holds for n = k + 1 whenever n = k holds.

Therefore, by method of Induction, we can conclude that n^2 ≥ n for all n ≥ 1

Note : We have only solved it for n as a Natural number. However you can try proving it for any real number.