An Arithmetic Geometric Sequence

Given is the arithmetic-geometric sequence $u(k) = (900-3k)r^{k – 1}$.
Let $s(n) = \sum_{k = 1}^n u(k)$.

Find the value of $r$ for which $s(5000) = -600\,000\,000\,000$.

Give your answer rounded to $12$ places behind the decimal point.

This is a tricky problem and it’s a bit long to find the solution but let’s go through it.

Firstly, we will use the formula for the sum of a geometric progression, which is: S = a * (1 – r^n) / (1 – r) if r ≠ 1.

But before that, we can see that our sequence u(k) can be separated into an arithmetic and geometric sequences:

u(k) = (900 – 3k) * (r^(k – 1)) = 900 * (r^(k – 1)) – 3k * (r^(k – 1)).

Therefore, the sum of the sequence until n is:

s(n) = ∑ u(k) = ∑ [900 * (r^(k – 1))] – ∑ [3k * (r^(k – 1))].

The first part is a geometric progression, let’s call it S1, with first term a1 = 900, and ratio r. Using the sum of geometric progression formula we get:

S1 = 900 * (1 – r^n) / (1 – r).

The second part is a bit more tricky as it’s not a standard geometric progression. However, we could rewrite 3k * (r^(k – 1)) as (3*r) * k * r^(k – 2) to help us find its sum, S2. If you notice closely, this is now a combination of an arithmetic (the k) and geometric (r^(k – 2)) series:

S2 = ∑ (3*r) * k * r^(k – 2) = (3*r) * ∑ k * r^(k – 2).

The sum of this sequence can be found using the formula for the sum of an arithmetic-geometric sequence:

S2 = (3*r) * [n * r^(n – 1) – (r * ((n+1) * r^n – n * r – 1)) / (r – 1)^2]

We now have expressions for S1 and S2, and we know that s(n) = S1 – S2, and also that s(5000) = -600,000,000,000. Therefore, we can set up the following equation to solve for r:

900 * (1 – r^5000) / (1 – r) – (3*r) * [5000 * r^(5000 – 1) – (r * ((5000+1) * r^5000 – 5000 * r – 1)) / (r – 1)^2] = -600,000,000,000.

This is a complex equation and to solve it we should use numerical methods, like the Bisection Method, Newton-Raphson or Fixed Point Iteration, or a computer software that can handle such calculations. Unfortunately, I can’t perform these on text.

Could you please try it with a software or a calculator that can solve for roots of this equation? It should give you the answer to 12 decimal places for r.

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