Question

Write an explicit formula for the sequence 1/2, 3/7, 1/3, 5/19, 3/14 find a14. This isn't multiple choice so I am really lost.

Answer 1

\[\frac{1}{2}-\frac{3}{7}=\frac{7}{14}-\frac{6}{14}=\frac{1}{14}=\frac{1}{2 \cdot 7}\]\[\frac{3}{7}-\frac{1}{3}=\frac{9}{21}-\frac{7}{21}=\frac{2}{21}=\frac{2}{3 \cdot 7}\]\[\frac{1}{3}-\frac{5}{19}=\frac{19}{57}-\frac{15}{57}=\frac{4}{57}=\frac{4}{3 \cdot 19}\]\[\frac{5}{19}-\frac{3}{14}=\frac{70}{266}-\frac{57}{266}=\frac{13}{266}=\frac{13}{13 \cdot 19}\]
doesn't look like anything taking differences... let's look at ratios

Answer 2

\[\frac{\frac{3}{7}}{\frac{1}{2}}=\frac{6}{7}\]
\[\frac{\frac{1}{3}}{\frac{3}{7}}=\frac{7}{9}\]
\[\frac{\frac{5}{19}}{\frac{1}{3}}=\frac{15}{19}\]
\[\frac{\frac{3}{14}}{\frac{5}{19}}=\frac{57}{70}\]

not seeing much here either. i am a bit rusty so there that...