# Stack Expansion

Computer Science Level 4

In an array-based implementation, whenever a stacks become full, it is common to just double its size. Suppose that instead of doubling every-time, we start with an array of size $$\alpha$$ and increase the array size by the sequence $2\alpha , 3\alpha , 4\alpha ...$ for some positive constant $$\alpha$$.

In such a stack, suppose we execute $$n$$ push operations. The total cost complexity $$g(n,\alpha)$$ of this operation can be asymptotically expressed as a function $$g$$ in terms of $$n$$ and $$\alpha$$. If $g(n,\alpha) = O( \large{ n^x \alpha^{y} )}$ for some exponents $$x$$ and $$y$$ . What is the value of $$10x^2 y^2$$?

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