How to find the greatest absolute term in a binomial expansion

Question

I know that the ratio of consecutive terms should be ≥1 and I'm able to solve using that approach but I was wondering whether we could derive a general formula and how would it work?

Answer 1

My instantaneous thought on this is that the first binomial coefficient proceeds from the zeroth (1) by multiplying it by $n$; and the second by multiplying that by $(n-1)/2$, ... , & thereafter by $(n-k)/(k+1)$; and you're also multiplying by $x$ each time ... so the maximum term will be at the stationary point, when $(n-k)x=k+1$, and the increase 'turns round' to become a decrease. So you would have $$k≈\frac{nx-1}{x+1} ,$$ the "≈" taking account of the fact that it goes in integer steps, & $x$ need not be an integer.