Question

1. insert three harmonic means between 2/5 and 4. 3. find the sum of the first 7 terms of the harmonic sequence: -5/21, -5/13, 1,...

Explanation:

Step1: Find the arithmetic sequence

If \(a = \frac{2}{5}\) and \(b = 4\), then the reciprocals \(A=\frac{5}{2}\) and \(B=\frac{1}{4}\). For an arithmetic sequence \(A,a_2,a_3,a_4,B\) with \(n = 5\) terms. The common difference \(d=\frac{B - A}{n - 1}=\frac{\frac{1}{4}-\frac{5}{2}}{4}=\frac{\frac{1 - 10}{4}}{4}=\frac{-9}{16}\)

Step2: Find the terms of the arithmetic sequence

\(a_2=A + d=\frac{5}{2}-\frac{9}{16}=\frac{40 - 9}{16}=\frac{31}{16}\)
\(a_3=A + 2d=\frac{5}{2}-\frac{9}{8}=\frac{20 - 9}{8}=\frac{11}{8}\)
\(a_4=A+3d=\frac{5}{2}-\frac{27}{16}=\frac{40 - 27}{16}=\frac{13}{16}\)

Step3: Find the harmonic means

The harmonic means are \(\frac{16}{31},\frac{8}{11},\frac{16}{13}\)

Answer:

The three harmonic means between \(\frac{2}{5}\) and \(4\) are \(\frac{16}{31},\frac{8}{11},\frac{16}{13}\)