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# If A, B, C and Dare in Continued Proportion, Then Prove that - ICSE Class 10 - Mathematics

#### 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"

#### Solution

"a"/"b" = "b"/"c" = "c"/"d" = "k"

⇒ c = kd

b =kc= k2

a= kb= k3

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)

Is there an error in this question or solution?

#### APPEARS IN

Frank Solution for Frank Class 10 Mathematics Part 2 (2016 to Current)
Chapter 9: Ratio and Proportion
Exercise 9.3 | Q: 10.2

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Solution If A, B, C and Dare in Continued Proportion, Then Prove that Concept: Proportions.
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