|If p, q, r, s, t are in continued proportion, show that = |
Sol: Given p, q, r, s, t are in continued proportion.
By definition ---- (i)
can be written as
i.e., when there are 5 terms in continued proportion, the ratio of the first to the last term is equal to the ratio of the fourth power of first term to the fourth power of the second.
Quantities are said to in continued proportion when the first is to the second, as the second is to the third, as the third is to the fourth, . . .
i.e, a, b, c, d, . . . are in continued proportion when
a/b = b/c = c/d = . . .
If three quantities a, b, c are in continued proportion, we have
a : b = b : c
⇒ ac = b2
In the above, b is called mean proportional between a and c;
and c is called third proportional to a and b.
If three quantities are proportionals, the first is to the third is the duplicate ratio of the first to the second
i.e, if a, b, c are proportionals,
a : c = a2 : b2