Question

Write an explicit formula for the sequence 1/2, 3/7, 1/3, 5/19, 3/14 Then find a[14]

Answer 1

\[\frac{1}{2},\frac{3}{7}, \frac{1}{3}, \frac{5}{19}, \frac{3}{14},\cdots\\=\frac{2}{4},\frac{3}{7}, \frac{4}{12}, \frac{5}{19}, \frac{6}{28},\cdots\]

the numerators form an arithmetic sequence

the denominators form a sequence that is quadratic, i.e., of the form \(an^2+bn+c\)

Answer 2

you can find a[14] by following the pattern in the numerators and the pattern in the denominators, without solving the quadratic expression "\(an^2+bn+c\)" that describe the pattern in the denominators