Introduction to the Problem

The evaluation of recursive sequences and limits is a fundamental aspect of advanced mathematics. This article explores a fascinating problem involving a sum of square roots, shedding light on the intricate relationships between algebraic expressions and their limiting values.

Defining the Sequences

We define the sequences (a_n) and (b_n) as follows:

The sum of these sequences from 1 to 99 is denoted by (a) and (b), respectively. Additionally, a third sequence (c_n sqrt{10sqrt{100-n}}) is introduced.

The Main Objectives

The primary goal is to evaluate the ratio:

( frac{sum_{n1}^{99} a_n}{sum_{n1}^{99} b_n} )

Key Observations and Calculations

First, observe that:

(sum_{n1}^{99} c_n sum_{n1}^{99} a_n )

Next, the ratio is simplified using the product of (a_n) and (b_n):

( frac{a}{b} frac{sum_{n1}^{99} a_n}{sum_{n1}^{99} b_n} frac{sqrt{10sqrt{1}}sqrt{10sqrt{2}}cdotssqrt{10sqrt{99}}}{sqrt{10-sqrt{1}}sqrt{10-sqrt{2}}cdotssqrt{10-sqrt{99}}} )

The product of (a_n) and (b_n) can be simplified as:

( a_n b_n^2 202sqrt{100-n} 210sqrt{100-n} 2c_n^2 )

This implies:

( a_n b_n sqrt{2}c_n )

Summing from 1 to 99:

( ab sqrt{2} sum_{n1}^{99} c_n sqrt{2}a )

Therefore:

( b sqrt{2} - a )

This leads to:

( frac{a}{b} frac{1}{sqrt{2}-1} sqrt{2} 1 )

Step-by-Step Solution

Let's define the term (Delta_n sqrt{10sqrt{n}} - sqrt{10-sqrt{n}}). Squaring this term and simplifying, we obtain:

( Delta_n sqrt{2} sqrt{10 - sqrt{100-n}} )

Summing all these terms from 1 to 99:

( sum_{n1}^{99} Delta_n sqrt{2} sum_{n1}^{99} sqrt{10 - sqrt{n}} sqrt{2} cdot b )

This implies:

( frac{a-b}{b} frac{a}{b} - 1 sqrt{2} )

Therefore, the final ratio:

( frac{a}{b} sqrt{2} 1 )

Conclusion

The value of the recursive sum ( frac{sum_{n1}^{99} sqrt{10sqrt{n}}}{sum_{n1}^{99} sqrt{10-sqrt{n}}} ) is ( sqrt{2} 1 ). This problem showcases the elegance and complexity of algebraic sequences and limits, providing a deeper understanding of mathematical relationships and recursive summations.