Prove the inequality for the indicated integer values of n 2

Prove the inequality for the indicated integer values of n, 2n^2 > (n + 1)^2, n greaterthanorequalto 3

Solution

2n^2 > (n+1)^2

2n^2 > n^2 +1 +2n

n^2 - 2n -1 >0

solve the inequality :

we get n < 1- sqrt2 and n > 1+sqrt2

Neglect n < 1- sqrt2 ; n < -0.4141

Now n> 1+sqrt2 ---> n > 2.414     ( sqrt2 = 1.414)

So, imediate highets integer to 2.414 is 3

So, given inequality is true for integers 3 and above

So, n >=3

Henec proved