Question

Expand:
(i) $(n - 1)(n + 1)(n^{2} + 1)$
(ii) $(n - \frac{1}{n} )(n + \frac{1}{n} )(n^{2} + \frac{1}{n^{2} } )$
(iii)$(x - 1)(x + 1)(x^{2} + 1)(x^{4} + 1)$
(iv) $(2x - y)(2x + y)(4x^{2} + y^{2})$.

Answer 1

Proving 1n+1+1n+2+⋯+12n>1324 for n>1,n∈N

To solve it I used induction but it is leading me nowhere my attempt was as follows:

Lets assume the inequality is true for n=k then we need to prove that it is true for k+1

so we need to prove 1k+2+1k+3+⋯+12(k+1)>13/24

I don't know where to go from here please help.