Advertisements
Advertisements
Question
If a, b, c and dare in continued proportion, then prove that
`sqrt (("a + b + c")("b + c + d")) = sqrt "ab" + sqrt "bc" + sqrt "cd"`
Advertisements
Solution
`"a"/"b" = "b"/"c" = "c"/"d" = "k"`
⇒ c = kd
b =kc= k2d
a= kb= k3d
`sqrt (("a + b + c")("b + c + d")) = sqrt "ab" + sqrt "bc" + sqrt "cd"`
LHS
`sqrt (("a + b + c")("b + c + d"))`
`= sqrt (("k"^3"d" + "k"^2"d" + "kd")("k"^2"d" + "kd" + "d"))`
`= sqrt ("kd" ("k"^2 + "k" + 1) xx "d"("k"^2 + "k" + 1))`
`= sqrt ("kd"^2 ("k"^2 + "k" + 1)^2)`
`= "d" sqrt "k" ("k"^2 + "k" + 1)`
RHS
= `sqrt "ab" + sqrt "bc" + sqrt "cd"`
= `sqrt ("k"^3"d" xx "k"^2"d") + sqrt ("k"^2"d" xx "kd") + sqrt ("kd" xx "d")`
=`sqrt ("k"^5"d"^2) + sqrt ("k"^3"d"^2) + sqrt "k" "d"^2`
= `"k"^2"d" sqrt "k" + "kd" sqrt "k" + "d" sqrt "k"`
= `"d" sqrt "k" ("k"^2 + "k" + 1)`
APPEARS IN
RELATED QUESTIONS
If ( a+c) : b = 5 : 1 and (bc + cd) : bd = 5 : 1, then prove that a : b = c : d
If `a/b = c/d` Show that a + b : c + d = `sqrt(a^2 + b^2) : sqrt(c^2 + d^2)`.
Find the fourth proportional to `(1)/(3), (1)/(4), (1)/(5)`
Determine if the following numbers are in proportion:
150, 200, 250, 300
Find the value of x if 5 : 3 : : x : 6.
If a, b, c, d are in continued proportion, prove that: (a + d)(b + c) – (a + c)(b + d) = (b – c)2
If a, b, c, d, e are in continued proportion, prove that: a : e = a4 : b4.
Determine if the following are in proportion.
15, 45, 40, 120
What number must be added to each of the numbers 4, 6, 8, 11 in order to get the four numbers in proportion?
What is the term "d" called in the expression a : b :: c : d?
