Insert 6

Insert four harmonic means between 3/7 and 3/19

Correct result:

a₂ = 0.355
a₃ = 0.303
a₄ = 0.2643
a₅ = 0.2344

Solution:

a_{1} = 3 / 7 = \frac{3}{7} ≐ 0.4286 a_{6} = 4 / 19 = \frac{4}{19} ≐ 0.2105 b_{1} = 1 / a_{1} = 1 / 0.4286 = \frac{7}{3} ≐ 2.3333 b_{6} = 1 / a_{6} = 1 / 0.2105 = \frac{19}{4} = 4.75 d = (b_{6} - b_{1}) / (6 - 1) = (4.75 - 2.3333) / (6 - 1) = \frac{29}{60} ≐ 0.4833 b_{2} = b_{1} + d = 2.3333 + 0.4833 = \frac{169}{60} ≐ 2.8167 a_{2} = 1 / b_{2} = 1 / 2.8167 = \frac{60}{169} = 0.355

b_{3} = b_{2} + d = 2.8167 + 0.4833 = \frac{33}{10} = 3.3 a_{3} = 1 / b_{3} = 1 / 3.3 = \frac{10}{33} = 0.303

b_{4} = b_{3} + d = 3.3 + 0.4833 = \frac{227}{60} ≐ 3.7833 a_{4} = 1 / b_{4} = 1 / 3.7833 = \frac{60}{227} = 0.2643

b_{5} = b_{4} + d = 3.7833 + 0.4833 = \frac{64}{15} ≐ 4.2667 a_{5} = 1 / b_{5} = 1 / 4.2667 = \frac{15}{64} = 0.2344

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