Continued Proportional | Examples on Continued Proportional |

Continued Proportional |
Examples on Continued Proportional |
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#Continued proportion
DEFINITION
Continued proportion
If a,b and c are in continued-proportion,
that means a:b::b:c
⇒ Product of extremes = Product of means
⇒a×c=b×b=b^2

Example : 8:4:2 is in continued proportion.

DEFINITION
#Continued Proportion
If a:b=b:c then a,b,c are said to be in continued proportion.
Here b is known as mean proportional.
Consider a continued proportion,
a:b=b:c
∴ b/a = c/b
∴a×c=b×b
∴b^2=ac
∴b= (ac)^2
Thus, we conclude that if a,b,c are in continued proportional then,
b= (ac)^2
#Continued Proportion
When quantities are in continued proportion, all the ratios are equal. If
a:b = b:c = c:d = d:e,
the ratio of a:b is the same, as that of b:c, of c:d, or of d:e. The ratio of the first of these quantities to the last, is equal to the product of all the intervening ratios; (Art. 348,) that is, the ratio of a:e is equal to a/b.b/c.c/d.d/e
But as the intervening ratios are all equal, instead of multiplying them into each other, we may multiply any one of them into itself; observing to make the number of factors equal to the number of intervening ratios. Thus the ratio of a:e, in the example just given, is equal to
​a/b.a/b.a/b.a/b=a^4/b^4
When several quantities are in continued proportion, the number of couplets, and of course the number of ratios, is one less than the number of quantities. Thus the five proportional quantities a, b, c, d, e, form four couplets containing four ratios; and the ratio of a:e is equal to the ratio of a4:b4, that is, the ratio of the fourth power of the first quantity, to the fourth power of the second. Hence,
If three quantities are proportional, the first is to the third, as the square of the first, to the square of the second; or as the square of the second, to the square of the third. In other words, the first has to the third, a duplicate ratio of the first to the second. And conversely, if the first of the three quantities is to the third, as the square of the first to the square of the second, the three quantities are proportional.
If a:b = b:c, then a:c = a^2:b^2.

[nwazali]

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