Harmonic Progression
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Unequal numbers :math:`a_1, a_2, a_3 ...` are said to be in H. P. if :math:`\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.

:math:`nth~term~of~an~H.~P. = \frac{1}{Corresponsing~term~in~corresponding~A.~P.}`

If :math:`a` is first term and :math:`b` is nth term then c. d.
:math:`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.

Problems
========
1. Find the 100th term of the sequence :math:`1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, ...`

2. If pth term of an H. P. be :math:`qr` and qth term be :math:`rp`, prove that rth term is :math:`pq`.

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

   :math:`(q - r)bc + (r - p)ca + (p - q)ab = 0`

4. If :math:`a, b, c` are in H. P. show that :math:`\frac{a - b}{b - c} = \frac{a}{c}`

5. If :math:`a, b, c, d` are in H. P.; then prove that :math:`ab + bc + cd = 3ad`

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

7. If :math:`a, b, c` are in H. P., show that

   :math:`\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}` are in H. P.

8. If :math:`a^2, b^2, c^2` are in A. P. show that :math:`b + c, c + a, a + b` are in A. P.

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

10. Find the 8th term of the sequence :math:`\frac{2}{11}, \frac{1}{5}, \frac{2}{9}, ...`

11. Find the 7th term of the series :math:`\frac{1}{3}, \frac{8}{23}, \frac{4}{11}, ...`

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

13. If mth term of an H. P. be :math:`n` and nth term be :math:`m`, prove that

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

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

15. If :math:`a, b, c` be in H. P., prove that

    :math:`\frac{1}{b - a} + \frac{1}{b - c} = \frac{1}{a} + \frac{1}{c}`

16. If :math:`a, b, c` be in H. P., prove that

    :math:`\frac{b + a}{b - a} + \frac{b + c}{b - c} = 2`

17. If :math:`x_1, x_2, x_3, x_4, x_5` are in H. P., prove that

    :math:`x_1x_2 + x_2x_3 + x_3x_4 + x_4x_5 = 4x_1x_5`

18. If :math:`x_1, x_2, x_3, x_4` are in H. P., prove that

    :math:`(x_1 - x_3)(x_2 - x_4) = 4(x_1 - x_2)(x_3 - x_4)`

19. If :math:`b + c, c + a, a + b` are in H. P., show that

    :math:`\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}` are in A. P.

20. If :math:`b + c, c + a, a + b` are in H. P., show that

    :math:`a^2, b^2, c^2` are in A. P.

21. If :math:`a, b, c` be in A. P., prove that

    :math:`\frac{bc}{ab + ac}, \frac{ca}{bc + ab}, \frac{ab}{ca + cb}` are in H. P.

22. If :math:`a, b, c` are in H. P., prove that

    :math:`\frac{a}{b + c - a}, \frac{b}{c + a - b}, \frac{c}{a + b - c}` are in H. P.

23. If :math:`a, b, c` are in H. P., prove that

    :math:`\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}` are in H. P.