# 41. 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.

## 41.1. 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.