A sequence of rational numbers is described as follows: (1)/(1), (3)/(2), (7)/(5), (17)/(12), …,

step 1 Okay, so this is our question number 110. The numerator series 13717, the n term is A plus 2B, a

A sequence of rational numbers is described as follows:...

A sequence of rational numbers is described as follows: $$ \frac{1}{1}, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \dots, \frac{a}{b}, \frac{a+2 b}{a+b}, \dots $$ Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let $x_{n}$ and $y_{n}$ be, respectively, the numerator and the denominator of the $n$ th fraction $r_{n}=x_{n} / y_{n}$ . a. Verify that $x_{1}^{2}-2 y_{1}^{2}=-1, x_{2}^{2}-2 y_{2}^{2}=+1$ and, more generally, that if $a^{2}-2 b^{2}=-1$ or $+1,$ then $(a+2 b)^{2}-2(a+b)^{2}=+1 \quad$ or $-1$ respectively. b. The fractions $r_{n}=x_{n} / y_{n}$ approach a limit as $n$ increases. What is that limit? (Hint: Use part (a) to show that $r_{n}^{2}-2=\pm\left(1 / y_{n}\right)^{2}$ and that $y_{n}$ is not less than $n$ .

A sequence of rational numbers is described as follows:
$$
\frac{1}{1}, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \dots, \frac{a}{b}, \frac{a+2 b}{a+b}, \dots
$$
Here the numerators form one sequence, the denominators form
a second sequence, and their ratios form a third sequence. Let $x_{n}$
and $y_{n}$ be, respectively, the numerator and the denominator of the
$n$ th fraction $r_{n}=x_{n} / y_{n}$ .
a. Verify that $x_{1}^{2}-2 y_{1}^{2}=-1, x_{2}^{2}-2 y_{2}^{2}=+1$ and, more
generally, that if $a^{2}-2 b^{2}=-1$ or $+1,$ then
$(a+2 b)^{2}-2(a+b)^{2}=+1 \quad$ or $-1$
respectively.
b. The fractions $r_{n}=x_{n} / y_{n}$ approach a limit as $n$ increases.
What is that limit? (Hint: Use part (a) to show that
$r_{n}^{2}-2=\pm\left(1 / y_{n}\right)^{2}$ and that $y_{n}$ is not less than $n$ .

Key Concepts

Recurrence Relations

Recurrence relations are equations that recursively define a sequence: each term is formulated as a function of its preceding terms. In this problem, the recurrence relation governing the numerators and denominators not only generates the sequence but also establishes the mathematical structure that ensures a systematic and predictable convergence towards the irrational limit.

Convergence of Sequences

Understanding convergence is central to analyzing how a sequence of numbers behaves as its index increases. Here, by showing that certain error terms diminish (for example, through properties like r_n² - 2 being inversely related to the square of the denominator), one can rigorously prove that the sequence of rational approximations converges to ?2.

Rational Approximations to Irrational Numbers

This concept involves constructing sequences of fractions that converge to an irrational number. Often achieved via methods such as continued fractions or solutions to Pell’s equation, these approximants illustrate how the density of rational numbers allows one to closely approach irrational limits, even though no rational number equals an irrational number.

Pell's Equation

Pell’s equation, typically written as x² - D·y² = ±1 (with D a non-square integer), is a classical quadratic Diophantine equation whose solutions provide excellent rational approximations to ?D. In the context of this problem with D = 2, the equation x² - 2y² = ±1 underpins the structure of the sequence, ensuring that each rational approximant (formed by the numerator and denominator sequences) relates in a way that approximates ?2 increasingly well.

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