How Many Recursions Does T(n) Need for k?

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The discussion centers on determining how many recursions T(n) requires to reach a value k, given the recursive definition T(0) = 1 and T(n) = T(n-1) + sqrt(T(n-1)). A suggestion is made to bound the growth of T(n) using a continuous function derived from solving the differential equation dy/dx = sqrt(y) with the initial condition y(0)=1, which indicates that T(n) grows slower than this function. The solution to the differential equation suggests that the constant c is 2, providing a useful bound for m, the number of recursions needed. Clarity on terminology is also noted, specifically regarding the use of "root" to refer to the square root function.
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T(0) = 1
T(n) = T(n-1) + root(T(n-1))

how many recursion does T(n) need to grow to the number k?
can I get this? root(k) < m < c root(k)
c is constant and m is the times we need for T(n) goes to k.

Any help appreciated!
 
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You can bound it with a continuous function.

Try solving dy/dx = sqrt(y) with y(0)=1, you can show that this function is bigger than T but gives a pretty good bound. Solving the above gives c=2 as the constant you're looking for.

BTW: Maybe I'm just dumb but I had to read you post about three times before I figured out that by "root" you meant square root or sqrt or \sqrt(.).
 
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