I would also suggest to search at approach0.xyz/search – there is a good chance that this has been asked and answered before.

Which step is not clear @PhysicsGuy or do you have any problem in proceeding after that?

I am not supposed to answer problem statement questions. You need to edit your question to include what you have done, where you are stuck, all using mathjax formatting, otherwise there is good chance that this question will be closed.

@AmanKushwaha Could you provide me a hint how to prove after that. I wrote in the question that I wasn't able to approach this question.

Expand LHS and then use AM$\geq$HM inequality on the $n$ terms you'll get @PhysicsGuy

Ok thanks. I will try and come back. @AmanKushwaha

I have a doubt. How will I expand the LHS? Should I open the sigma notation? @AmanKushwaha

Choose any $j,k$ with $j<k\le n.$ Let $a_i$ remain fixed for $j\ne i\ne k$. Let $a_j$ and $a_k$ vary, but subject to the condition that $a_j+a_k$ remains constant. So $S$ also remains constant. Show the LHS is minimized when $a_j=a_k$. Deduce from this that, for a given $S,$ the LHS has a minimum when all the $a_i$ are equal. But when all the $a_i$ are equal, the LHS is $n^2/(n-1).$

@DanielWainfleet thanks

← 77 messages moved from Discussion between Aman Kushwaha and PhysicsGuy