Provide an appropriate response.
-A sequence of rational numbers $\left\{ \mathrm { r } _ { \mathrm { n } } \right\}$ is defined $b \mathrm { y } \mathrm {} \mathrm { r } _ { 1 } = \frac { 2 } { 1 }$, and if $\mathrm { r } _ { \mathrm { n } } = \frac { \mathrm { a } } { \mathrm { b } }$ then $\mathrm { r } _ { \mathrm { n } } + 1 = \frac { \mathrm { a } + \mathrm { b } } { \mathrm { a } - \mathrm { b } }$. Find $\mathrm { r } _ { 50 }$.
A) 3
B) 49
C) 50
D) 2

Question

Quizplus · Accepted Answer

The given sequence starts with $r_1 = \frac{2}{1}$. The rule for generating the next term is to add 1 to the current term, which can be rewritten as $\frac{a + b}{b}$ for the current term $\frac{a}{b}$, and then transform it according to the given formula to get the next term as $\frac{a + b}{a - b}$. Applying this rule repeatedly, it can be observed that each term simplifies to an integer, with the sequence starting as 2, 3, 4, 5, ..., indicating that each term is simply one more than the previous term. Therefore, $r_{50}$ is 51, but since this option is not available and there seems to be a misunderstanding in interpreting the sequence or the options provided, the closest correct interpretation based on the initial sequence and rule provided would be to consider the pattern that emerges from the application of the rule, which might have been intended to illustrate a different sequence progression or an error in the question's formulation.

Provide an Appropriate Response {rn}\left\{ \mathrm { r } _ { \mathrm { n } } \right\}{rn​}

Provide an Appropriate Response $\left\{ \mathrm { r } _ { \mathrm { n } } \right\}$