Let $S_1$ denote the sequence of positive integers $1, 2 ,3 ,4, 5, 6,...$ and define the sequence $S_{n+1}$ in terms of $S_n$ by adding 1 to those integers in $S_n$ which are divisible by $n$. Thu, for example $S_2$ is $2, 3, 4, 5, 6, 7, ..., S_3$ is $3,3,5,5,7,7...$ Determine those integers n with the property that be first $n - 1$ integers in $S_n$ are $n$
I am not sure if this helps without problems: Let $S_n$ be the sum of the first n terms of the sequence ${{1/2^n}}$. From the above, we see that $S_1=\frac{1}{2}, S_2 = \frac{3}{4}$, etc. Our formula at the end shows that $S_n= 1−\frac{1}{2^n}$.
Or calculate the first ones that work
Hint:
Term $n-1$ in $S_2$ has the value $n$. If term $n-1$ in $S_n$ still has the value $n$ then this means this value $n$ was not changed at any point during the construction of $S_3, S_4, \dots S_n$ (because once a value in the sequence is changed, it can only increase - it can never go back to its original value).
But if $n$ was divisible by some $m$ in the range $2 \le m \le n-1$ then the value $n$ would have been replaced by $n+1$ in $S_{m+1}$ ...