Analyze a mathematical sequence of fractions, examining their primality and frequency characteristics | Step-by-Step Solution

What We're Solving:

We need to analyze the sequence r_n = 2 + (n-1)/(n+1) by creating a table that shows each term as an irreducible fraction, then examine the primality of numerators and denominators, and look for frequency patterns.

The Approach:

This problem combines several mathematical concepts beautifully!

• Simplify algebraic fractions (algebra skills)
• Reduce fractions to lowest terms (number theory)
• Test for prime numbers (number theory)
• Look for patterns (mathematical reasoning)

Step-by-Step Solution:

Step 1: Understand the formula r_n = 2 + (n-1)/(n+1)

Rewrite this with a common denominator: r_n = 2(n+1)/(n+1) + (n-1)/(n+1) = (2n+2+n-1)/(n+1) = (3n+1)/(n+1)

Our sequence is r_n = (3n+1)/(n+1).

Step 2: Calculate the first several terms For n=1,2,3:

• n=1: r_1 = (3×1+1)/(1+1) = 4/2 = 2/1 (reduced)
• n=2: r_2 = (3×2+1)/(2+1) = 7/3 (already reduced since gcd(7,3)=1)
• n=3: r_3 = (3×3+1)/(3+1) = 10/4 = 5/2 (reduced)

Step 3: Check if fractions are in irreducible form For each fraction a/b, find gcd(a,b). If gcd(a,b)=1, it's already irreducible!

Step 4: Test primality For each numerator and denominator in the reduced fraction, check if it's prime:

• A prime number has exactly two factors: 1 and itself
• Remember: 1 is not prime, 2 is the only even prime

Step 5: Create your table Your table should have columns like: | n | r_n original | r_n reduced | Numerator prime? | Denominator prime? |

Step 6: Look for patterns After calculating several terms, ask:

• Do certain prime numbers appear more frequently?
• Is there a pattern in when numerators/denominators are prime?
• What happens as n gets larger?

The Framework:

Here's how to organize your work:

• 1. Computation Section: Show your algebraic simplification and calculate at least 10-15 terms
• 2. Data Table: Organize your results clearly with all requested information
• 3. Primality Analysis: Summarize which numbers appeared and their prime status
• 4. Frequency Analysis: Count how often primes appear in numerators vs denominators
• 5. Pattern Recognition: Describe any interesting patterns you notice
• 6. Conclusion: Summarize your findings

Memory Tip:

Remember that (3n+1)/(n+1) can be rewritten as 3 - 2/(n+1) using polynomial long division! This alternative form might help you see why the sequence approaches 3 as n gets large, and could reveal additional patterns in your analysis.

The key to success here is being systematic and patient - mathematics rewards careful, organized thinking!