Advertisements
Advertisements
Question
If a, b, c are in continued proportion, prove that: `(1)/a^3 + (1)/b^3 + (1)/c^3 = a/(b^2c^2) + b/(c^2a^2) + c/(a^2b^2)`
Advertisements
Solution
As a, b, c, are in continued proportion
Let `a/b = b/c` = k
L.H.S. = `(1)/a^3 + (1)/b^3 + (1)/c^3`
= `(1)/(ck^2)^3 + (1)/(ck)^3 + (1)/c^3`
= `(1)/(c^3k^6) + (1)/(c^3k^3) + (1)/c^3`
= `(1)/c^3[1/k^6 + 1/k^3 + 1/1]`
R.H.S. = `a/(b^2c^2) + b/(c^2a^2) + c/(a^2b^2)`
= `ck^2/((ck)^2c^2) + "ck"/(c^2(ck^2)^2) + c/((ck^2)^2(ck)^2)`
= `(ck^2)/(c^4k^2) + "ck"/(c^4k^4) + c/(c^4k^6)`
= `(1)/c^3 + (1)/(c^3k^3) + (1)/(c^3k^6)`
= `(1)/c^3[1 + 1/k^3 + 1/k^6]`
= `(1)/c^3[1/k^6 + 1/k^3 + 1]`
∴ L.H.S. = R.H.S.
APPEARS IN
RELATED QUESTIONS
If a, b, c are in continued proportion, show that: `(a^2 + b^2)/(b(a + c)) = (b(a + c))/(b^2 + c^2)`.
Using properties of proportion, solve for x:
`(sqrt(x + 5) + sqrt(x - 16))/(sqrt(x + 5) - sqrt(x - 16)) = 7/3`
Given four quantities p, q, r and s are in proportion, show that
q2(p - r) : rs (q - s) =(p2- q2- pq): ( r2-s2-rs).
If p, q and r in continued proportion, then prove the following:
(p2 - q2)(q2 + r2) = (q2 - r2)(p2 + q2)
If u, v, w, and x are in continued proportion, then prove that (2u+3x) : (3u+4x) : : (2u3+3v3) : (3u3+4v3)
`("pqr")^2 (1/"p"^4 + 1/"q"^4 + 1/"r"^4) = ("p"^4 + "q"^4 + "r"^4)/"q"^2`
What number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional?
Verify the following:
60 : 105 : : 84 : 147
Two numbers are in the ratio 5 : 7 and the sum of these numbers is 252. The larger of these numbers is
If a, b, c, d are in continued proportion, prove that: (a + d)(b + c) – (a + c)(b + d) = (b – c)2
Determine if the following are in proportion.
33, 44, 75, 100
