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Harmonic Progression

August 17, 2020
Category: Harmonic Progression

☼ Harmonic Progression ( H.P.) : The progression $a_1,a_2,a_3,...$ is called an H. P., if $\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},...$ is an A. P.

In this case, $b=\frac{2ac}{a+c}$

(i) If a, b, c, d, e, f, . . are in H. P., then

$\left (\frac{1}{b}-\frac{1}{a} \right )= \left (\frac{1}{c}-\frac{1}{b} \right )=\left (\frac{1}{d}-\frac{1}{c} \right )=\left (\frac{1}{e}-\frac{1}{d} \right ) =\left (\frac{1}{f}-\frac{1}{e} \right )$

(ii) If a, b, c are in H. P., then b is called the harmonic mean ( H.M.) between a and c. In this case, $b=\frac{2ac}{a+c}$

e.g.

(1) A car covers a certain distance at 75 km / hr. and returns back to the starting point at 50 km / hr. Find his average speed during the whole journey.

Average Speed $= \frac{2xy}{x+y}$

$= \frac{2\times75\times50}{75+50}$

= 60 km / hr.

(2) A man covers a certain distance at 60 km / hr. and returns back to the starting point at 40 km / hr. Find his average speed during the whole journey.

( Try yourself ) Ans. : 48 km / hr

(iii) Harmonic mean of $a_{1},a_{2},a_{3},...,a_{n},$ is = H. M. = $\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}}$

e.g.

(1) A bus travels along the four sides of a square at the speeds of 800, 600, 400 and 200 km / hr. Find the average speed of the bus around the field.

Average Speed = $\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}}$

Average Speed = $\frac{4}{\frac{1}{800}+\frac{1}{600}+\frac{1}{400}+\frac{1}{200}}$

$= \frac{4}{\frac{3+4+6+12}{2400}}$

$= 4\times\frac{2400}{25}$

$= 384$ km/hr.

(2) An aeroplane flies along the three sides of an equilateral triangle at the speeds of 200, 600 and 1200 km / hr. Find the average speed of the bus around the field.

( Try yourself ) Ans. : 400 km / hr.

If A.M., G.M. and H.M. of two nos. x and y are respectively A, G and H, then G = H · A

e.g. for A.M., G.M. and H.M. of . 2 and 8 are respectively A = 5 , G = 4 and H = 3.2, then

$G^{2}= H \times A$ ( i.e. $4^{2} = 3.2\times 5=16$ )