Solve the recurrence relation T(n) = c(T(n/c) + 1), T(1) = 1, by finding an expression for T(n) in big-Oh notation.
Think about inputs of the form $c^k$. $$T(c^k)=cT(c^{k-1})+c=c^2T(c^{k-2})+c^2+c=\;...= > c^kT(1)+c+c^2+c^3+...+c^k$$ $$T(c^k)=\frac{c\cdot > c^k-1}{c-1}+c^kT(1)$$
If we want this function to be continuous, we note that $$T(n)=T(c^{\log_cn})=\frac{cn-1}{c-1}+T(1)n$$
If we let $a=T(1)+\frac 1{c-1}$ be an arbitrary constant, we get $$T(n)=an-\frac1{c-1}$$
This is the general form of the recursion. To find the specific form given your constraint, we substitute $T(1)=1$ into our equation for $a$, getting $a=1+\frac 1{c-1}=\frac c{c-1}$. So, the final solution is $$T(n)=\frac{c\cdot n-1}{c-1}$$
And, this function has complexity of $O(n)$ as it is a linear function.
So this is a question that has been answered, but this post in particular is look for alternative methods, or an explanation for the solution that I already got.
The following solution by Don Thousand does a great job of going through the steps, but I'm a beginner at this and I don't understand the process or reasoning behind a lot of the work, and was hoping someone could help me understand this method, or possibly provide an alternative.
Assume a linear ansatz of $\ T(n) := a\ n + b.\ $ The recursion equation we now have is $$ T(n) = a\ n + b = c(T(n/c) + 1) = c( a\ n/c + b+1) = a\ n + b\ c + c. $$ Solve this equation to get $\ b = -c/(c-1). \ $ Thus we have $\ T(n) = a\ n - c/(c-1) \ $ and $\ T(n) = O(n)\ $ as we expected. We need to find $\ a\ $ using $\ T(1)=1 \ $ with solution $\ a = (2c-1)/(c-1).\ $ Finally, $\ T(n) = ((2c-1)n-c)/(c-1).$
The key to this solution is the ansatz which was a guess. It worked, but in general, it would be rare to be able to guess the exact form of the solution. Clearly we were lucky in this case.
In case the answer is supposed to be O(n^k), then the obvious guess is a polynomial of degree $k$ in $n$. In rare cases like this, that is all you need to do. The technique that you present if much more complicated using analysis and some transcendental functions. Also I does't match exactly my answer because all it cares about is $\ O(n).$