Question

Show that (1 + 2 +. . .+n)² > 1² +. . .+ n², for n ≥ 2.

Answer 1

As we can can see that the first term(left side term) is nothing but the square of the sum of first "N" natural number

which is nothing but is equals to

because the sum of first n natural number is (n*(n+1)/2).

so it will be equal to

Now the second term(right side term) is nothing but the sum of square of first "N" natural numbers And it is given by   

   Which is equal to

now we can write first term as ,

now as given "n" must be greater or equal to 2, let's check if

   is positive for minimum value of "n" or not because as the second term is contained in the first term as shown in above now if it is positive means after subtracting the second term from the first term we will get positive number that means first term is greater than the right side term. if we put n=2 in we get (16-8-2)/6 =1 which is [positive number that means the left hand side terms is greater than right hand side term.