34. Harmonic Progression

Unequal numbers \(a_1, a_2, a_3 ...\) are said to be in H. P. if \(\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, ...\) are in A. P. Thus, you can easily observe that no term in an H. P. can be 0 because that would make reciprocal infinite.

\(nth~term~of~an~H.~P. = \frac{1}{Corresponsing~term~in~corresponding~A.~P.}\)

If \(a\) is first term and \(b\) is nth term then c. d. \(d = \frac{\frac{1}{b} - \frac{1}{a}}{n - 1}\).

There are few properties of H. P. is there by itself bu rather we solve problems related to harmonic progressions by treating their terms’ reciprocal in A. P.

34.1. Problems

  1. Find the 100th term of the sequence \(1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, ...\)

  2. If pth term of an H. P. be \(qr\) and qth term be \(rp\), prove that rth term is \(pq\).

  3. If pth, qth and rth terms of an H. P. be respectively a, b and c then prove that

    \((q - r)bc + (r - p)ca + (p - q)ab = 0\)

  4. If \(a, b, c\) are in H. P. show that \(\frac{a - b}{b - c} = \frac{a}{c}\)

  5. If \(a, b, c, d\) are in H. P.; then prove that \(ab + bc + cd = 3ad\)

  6. If \(x_1, x_2, x_3, ..., x_n\) are in H. P., prove that \(x_1x_2 + x_2x_3 + x_3x_4 + ... + x_{n - 1}x_n = (n - 1)x_1x_n\)

  7. If \(a, b, c\) are in H. P., show that

    \(\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}\) are in H. P.

  8. If \(a^2, b^2, c^2\) are in A. P. show that \(b + c, c + a, a + b\) are in A. P.

  9. Find the sequence whose nth term is \(\frac{1}{3n - 2}\). Is this sequence in an H. P.?

  10. Find the 8th term of the sequence \(\frac{2}{11}, \frac{1}{5}, \frac{2}{9}, ...\)

  11. Find the 7th term of the series \(\frac{1}{3}, \frac{8}{23}, \frac{4}{11}, ...\)

  12. Find the 4th term of an H. P. whose 7th term is \(\frac{1}{20}\) and 13th term is \(\frac{1}{38}.\)

  13. If mth term of an H. P. be \(n\) and nth term be \(m\), prove that

    \((m + n)th term = \frac{mn}{m + n}\) and \((mn)th term = 1\)

  14. The sum of three rational numbers in H. P. is 37 and the sum of their reciprocals is \(\frac{1}{4}\); find the numbers.

  15. If \(a, b, c\) be in H. P., prove that

    \(\frac{1}{b - a} + \frac{1}{b - c} = \frac{1}{a} + \frac{1}{c}\)

  16. If \(a, b, c\) be in H. P., prove that

    \(\frac{b + a}{b - a} + \frac{b + c}{b - c} = 2\)

  17. If \(x_1, x_2, x_3, x_4, x_5\) are in H. P., prove that

    \(x_1x_2 + x_2x_3 + x_3x_4 + x_4x_5 = 4x_1x_5\)

  18. If \(x_1, x_2, x_3, x_4\) are in H. P., prove that

    \((x_1 - x_3)(x_2 - x_4) = 4(x_1 - x_2)(x_3 - x_4)\)

  19. If \(b + c, c + a, a + b\) are in H. P., show that

    \(\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}\) are in A. P.

  20. If \(b + c, c + a, a + b\) are in H. P., show that

    \(a^2, b^2, c^2\) are in A. P.

  21. If \(a, b, c\) be in A. P., prove that

    \(\frac{bc}{ab + ac}, \frac{ca}{bc + ab}, \frac{ab}{ca + cb}\) are in H. P.

  22. If \(a, b, c\) are in H. P., prove that

    \(\frac{a}{b + c - a}, \frac{b}{c + a - b}, \frac{c}{a + b - c}\) are in H. P.

  23. If \(a, b, c\) are in H. P., prove that

    \(\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}\) are in H. P.