# Summation: finding the upper bound algebraically

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Hello, Recently, I've encountered a rather unusual (for me) problem. Here's what it looks like: What is the minimum value of x for which this relationship is true? N ≤ A * Σ( (4b/5)^n, n=0, x-1 ) I need to find the minimum value of x, but notice that the x is found in the expression for the upper bound of the sum. I've never dealt with that before. Is there a way to solve this algebraically? The context of the problem gives some additional indications for the other variables. First, all variables are whole numbers. Second, the value of b is an integer between 0 and 8 (but no more than 8). If there is no way to solve this symbolically, then I suppose that I will be content with an estimate. However, my attempt to find a suitable estimate led me to another mathematical dead end. This year in my calculus class, we learned the integral test: Σ(ak, k=1, ∞) ≤ a1 + ∫(a(x)dx, 1, ∞) Hence, this must be true (right?): N ≤ A + ∫( ( 4b/5)^t, 0, x-1 ) (Because the original summation was a geometric sequence in the form ar^n where a = A, the first term a1 = A). From here, I antidifferentiated and substituted. Then I multiplied the last term of the resulting relation by (4b/5)/(4b/5) to get this: N ≤ A + ((4b/5)x-(4b/5)2)/((4b/5)*ln(4b/5)) I'm pretty sure my work is right... but where do I go from here? I predict that when I solve this equation, the result will be in the form of: x ≥ some very complex expression From that point, would it be admissible to simply plug in the known values of N, A, and b and round up to the nearest whole number (the "ceiling")? Would that yield a valid estimate? Surely there is a simpler way to solve (or approximate) this? Please help!

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A sum of the form Σk ak is called a geometric progression, and it has a pretty simple expression without the sum.

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I suppose you are referring to this (from the linked article):

Thanks for the help. I'll try to work with it and see what happens.

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