A sequence of rational numbers {rn} is defined by r1 = 2 /1 , and if rn = a/b then rn+1 = (a + b)/ (a - b) Find r50
Added by Taylor A.
Step 1
Now let's find the next few terms to see if we can find a pattern. r2 = (2 + 1) / (2 - 1) = 3/1 r3 = (3 + 1) / (3 - 1) = 4/2 = 2/1 Notice that r3 is the same as r1. Let's see if this pattern continues. r4 = (2 + 1) / (2 - 1) = 3/1 r5 = (3 + 1) / (3 - 1) = 4/2 Show more…
Show all steps
Close
Your feedback will help us improve your experience
Madhur L and 90 other Calculus 1 / AB educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
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 $n$ .
Infinite Sequences and Series
Sequences
If a and b are rational numbers, with b ≠ 0, and r is an irrational number, then a + b r is irrational.
Madhur L.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD