The harmonic imply (H) of n numbers ( x1, x2, x3, … , xn ), additionally referred to as subcontrary imply, is given by the system beneath.

If n is the variety of numbers, it’s discovered by dividing the variety of numbers by the reciprocal of every quantity.

What’s the system to seek out the harmonic imply of two or three numbers?

Suppose there are two numbers x1 and x2.

Suppose there are 3 numbers x1, x2, and x3.

H =

3
/
1/x1 + 1/x2 + 1/x3

Examples exhibiting methods to calculate the harmonic imply

Instance #1:

Discover the harmonic imply of three and 4

Instance #2:

Discover the harmonic imply of 1, 2, 4, and 10

H =

4
/
1/1 + 1/2 + 1/4 + 1/10

H =

4
/
20/20 + 10/20 + 5/20 + 2/20

A linear movement drawback that results in the harmonic system.

A automobile travels with a velocity of 40 miles per hour for the primary half of the way in which. Then, the automobile travels with a velocity of 60 miles per hour for the second half of the way in which. What’s the common velocity? 

Common velocity =

whole distance
/
whole time

First discover that it isn’t potential to make use of immediately the velocity system since we have no idea for a way lengthy the automobile stored driving with a velocity of 40 m/h after which 60 m/h. Nevertheless, with some manipulation, we are able to nonetheless deal with the issue.

Let t1 be the time it took to journey the primary half of the overall distance

Let d be the primary half of the overall distance.

Let t2 be the time it took to journey the second half of the overall distance

Let d be the second half of the overall distance.

Whole time = t1 + t2 = d/40 + d/60

Whole distance = d + d = second

Now exchange these within the system

Common velocity =

whole distance
/
whole time

Common velocity =

second
/
d/40 + d/60

Common velocity =

second
/
d(1/40 + 1/60)

Cancel d and the typical velocity =

2
/
(1/40 + 1/60)

Now, you possibly can see that it seems like we’re calculating the harmonic imply for two numbers by utilizing the system above.

H = common velocity =

2
/
(3/120 + 2/120)

H = common velocity =

2
/
(5/120)

H = common velocity =

2 × 120
/
5

H = common velocity =

240
/
5

= 48 miles per hour

One other solution to specific the harmonic imply of n numbers

That is going to problem you a bit. Nevertheless, don’t hand over. Preserve studying and you’re going to get it! Moreover, ensure you completely perceive fractions earlier than studying this part of the lesson.

Right here is our technique:

Step 1. Categorical the harmonic imply of two or three numbers another way.

Step 2. Look at rigorously step 1 by in search of patterns and make a generalization utilizing the summation symbols and the product symbols.

Rewriting the harmonic imply of two numbers

$$ H = frac{2}{ frac{1}{x_1} + frac{1}{x_2} } $$

$$ H = frac{2}{ frac{x_2 + x_1}{x_1 instances x_2} } $$

$$ H = frac{2 instances x_1x_2 }{ x_2 + x_1 } $$

$$ H = frac{2 instances x_1x_2 }{ frac{x_1x_2}{x_1} + frac{x_1x_2}{x_2} } $$

At this level, discover that we rewrote the denominator x2 + x1. Why did we do this? We did this as a result of we would like the x1x2 to seem in three totally different locations (as soon as on high and twice on the backside)

It will assist us to issue the underside a part of the advanced fraction as you possibly can see beneath.

$$ H = frac{2 instances x_1x_2 }{ x_1x_2(frac{1}{x_1} + frac{1}{x_2}) } $$

Rewriting the harmonic imply of three numbers

$$ H = frac{3}{ frac{1}{x_1} + frac{1}{x_2} + frac{1}{x_3} } $$

$$ H = frac{3}{ frac{x_2x_3 + x_1x_3 + x_1x_2}{x_1 instances x_2 instances x_3} } $$

$$ H = frac{3 instances x_1x_2x_3 }{ x_2x_3 + x_1x_3 + x_1x_2 } $$

$$ H = frac{3 instances x_1x_2x_3 }{ frac{x_1x_2x_3}{x_1} + frac{x_1x_2x_3}{x_2} + frac{x_1x_2x_3}{x_3} } $$

Discover once more that we rewrote the denominator x2x3 + x1x3 + x1x2. Why did we do this? We did this as a result of we would like the x1x2x3 to seem in 4 totally different locations (as soon as on high and thrice on the backside)

Once more, it will assist us to issue the underside a part of the advanced fraction as you possibly can see beneath.

$$ H = frac{3 instances x_1x_2x_3 }{ x_1x_2x_3(frac{1}{x_1} + frac{1}{x_2} + frac{1}{x_3}) } $$

Abstract

For two or 3 numbers, here’s what we’ve to this point!

$$ H = frac{2 instances x_1x_2 }{ x_1x_2(frac{1}{x_1} + frac{1}{x_2}) } $$

$$ H = frac{3 instances x_1x_2x_3 }{ x_1x_2x_3(frac{1}{x_1} + frac{1}{x_2} + frac{1}{x_3}) } $$

For n numbers, here’s what we may have then!

$$ H = frac{n instances x_1x_2x_3 … x_n }{ x_1x_2x_3 … x_n(frac{1}{x_1} + frac{1}{x_2} + frac{1}{x_3} +… + frac{1}{x_n}) } $$

For n numbers, we are able to make the system look somewhat higher or extra generalized by utilizing the summation image and the product image talked about earlier. Utilizing the product image, we get:

$$ H = frac{n instances prod_{j=1}^n x_j }{ prod_{j=1}^n x_j(frac{1}{x_1} + frac{1}{x_2} + frac{1}{x_3} +… + frac{1}{x_n}) } $$

And utilizing additionally the summation image, we get:

$$ H = frac{n instances prod_{j=1}^n x_j }{ prod_{j=1}^n x_j(sum_{i=1}^n frac{1}{x_i}) } $$

$$ H = frac{n instances prod_{j=1}^n x_j }{ (sum_{i=1}^n frac{prod_{j=1}^n x_j}{x_i}) } $$

See an instance of harmonic imply associated to the inventory market. 

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