Largest term in binomial expansion of $(1+n)^k$

Question

The binomial expansion of $(1+n)^k$ is

$$(1+n)^k=1+\binom{k}{1}n+\binom{k}{2}n^2+\cdots+\binom{k}{k}n^k.$$

If $n=1$, then the term in the middle is the largest, i.e. when $i=\lfloor k/2\rfloor$ and $i=\lceil k/2\rceil$.

What about for other integers $n$? Which term is the largest?

Answer 1

Hint. Prove that $$\binom k{i+1} n^{i+1}=\binom ki n^i\frac{n(k-i)}{i+1}$$ and then consider whether the RHS is larger than, equal to or smaller than $\binom kin^i$.

Answer 2

Most problems of this kind are quite easily resolved by considering whether the (i+1)th term is greater than the previous term, try and find whether it "monotonically" increases till one particular value of i and then "monotonically" decreases. The point where the change occurs is the maximum you're looking for. Similar reasoning gives you the minimum if it exists.