Solving the Sequence: Discovering the Missing Fraction

In mathematics, sequences often present intriguing challenges that require keen observation and analytical skills. Today, we will dive into solving a particular sequence to uncover the missing fraction. The sequence in question is:

frac{1}{2} frac{3}{3} frac{5}{5} frac{7}{7} ___ frac{11}{13} frac{13}{17}

Analysis of Numerators and Denominators

The task is to identify the missing fraction in the sequence. To do so, we will analyze the numerators and denominators separately.

Numerators

The numerators in the given sequence are:

1 1st odd number 3 2nd odd number 5 3rd odd number 7 4th odd number ___ 5th odd number (missing) 9 6th odd number 11 7th odd number

The pattern here is clear: the numerators are successive odd numbers. Starting from 1, the odd numbers are: 1, 3, 5, 7, 9, 11, ... Thus, the missing numerator in the sequence is:

9

Denominators

The denominators in the given sequence are:

2 1st prime number 3 2nd prime number 5 3rd prime number 7 4th prime number ___ 5th prime number (missing) 11 6th prime number 13 7th prime number

Here, the denominators are the sequence of prime numbers: 2, 3, 5, 7, 11, 13, 17. Therefore, the missing denominator in the sequence is:

11

Conclusion

By analyzing the numerators and denominators separately, we have identified the missing fraction in the sequence:

frac{9}{11}

Thus, the complete sequence is:

frac{1}{2} frac{3}{3} frac{5}{5} frac{7}{7} frac{9}{11} frac{11}{13} frac{13}{17}

Understanding and solving such sequences can enhance problem-solving skills in mathematics and are particularly relevant for students and individuals preparing for competitive exams.