How would you prove this question about tuples

Question

Let $n$ be a positive integer. Using induction, prove that there are $2^{(n−1)}$ tuples $(m_1, \ldots , m_k)$ of positive integers with the property that $m_1 + \cdots + m_k = n$.

Answer 1

Hint: $n = (m_1 + ....+m_k)$ can written $2^{n-1}$ ways,

Then $n+1 = 1+(m_1 + ....+m_k)$ can also be written in $2^{n-1}$ ways.

How many ways can $n+1 = 2+ (m_1 + ..... + m_k)$ be written?

What about $n+ 1 = n + (m_1+..... + m_k)$?

And what about for some $1 < j < n$, then how many ways can $n+1=j + (m_1 + ..... + m_k)$ be written?