Question

find the next two terms in this sequence.
\\(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{?}, \frac{square}{square}\\)

Explanation:

Step1: Analyze numerator pattern

The numerators are 1, 2, 3, 4, 5,... which follow the pattern of \(n\) where \(n\) starts at 1 and increases by 1 each term.

Step2: Analyze denominator pattern

The denominators are 2, 3, 4, 5,... which follow the pattern of \(n + 1\) where \(n\) is the numerator. So for the \(n\)-th term, the fraction is \(\frac{n}{n + 1}\).

Step3: Find the 6th term

For the 6th term, \(n = 6\), so the fraction is \(\frac{6}{6+1}=\frac{6}{7}\).

Step4: Find the 7th term

For the 7th term, \(n = 7\), so the fraction is \(\frac{7}{7 + 1}=\frac{7}{8}\).

Answer:

The next two terms are \(\frac{6}{7}\) and \(\frac{7}{8}\). So the sequence becomes \(\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},\frac{6}{7},\frac{7}{8}\) (first blank: 6, second blank: 7 for the fifth term's denominator; then the next term is \(\frac{6}{7}\) and then \(\frac{7}{8}\)).