Advertisements
Advertisements
Question
If x, y and z are in continued proportion, Prove that:
`x/(y^2.z^2) + y/(z^2.x^2) + z/(x^2.y^2) = 1/x^3 + 1/y^3 + 1/z^3`
Advertisements
Solution
Given: x, y and z are in continued proportion.
∴ `x/y = y/z`
⇒ y2 = xz
To prove: `x/(y^2.z^2) + y/(z^2.x^2) + z/(x^2.y^2) = 1/x^3 + 1/y^3 + 1/z^3`
Proof: Solving L.H.S.:
`x/(y^2.z^2) + y/(z^2.x^2) + z/(x^2.y^2)`
⇒ `(x^3 + y^3 + z^3)/(x^2.y^2.z^2)`
⇒ `(x^3 + y^3 + z^3)/(x^2.xz.z^2)`
⇒ `(x^3 + y^3 + z^3)/(x^3.z^3)`
⇒ `x^3/(x^3.z^3) + y^3/(x^3.z^3) + z^3/(x^3.z^3)`
⇒ `1/z^3 + y^3/(xz)^3 + 1/x^3`
⇒ `1/z^3 + y^3/(y^2)^3 + 1/x^3`
⇒ `1/z^3 + y^3/y^6 + 1/x^3`
⇒ `1/z^3 + 1/y^3 + 1/x^3`
Since L.H.S. = R.H.S.
Hence proved.
APPEARS IN
RELATED QUESTIONS
If y is the mean proportional between x and z, prove that: `(x^2 - y^2 + z^2)/(x^(-2) - y^(-2) + z^(-2)) = y^4`.
Given four quantities a, b, c and d are in proportion. Show that: (a – c)b2 : (b – d)cd = (a2 – b2 – ab) : (c2 – d2 – cd)
If `a/b = c/d` prove that each of the given ratios is equal to
`(5a + 4c)/(5b + 4d)`
Using properties of proportion, solve for x:
`(3x + sqrt(9x^2 - 5))/(3x - sqrt(9x^2 - 5)) = 5`
If a, b, c, d are in continued proportion, prove that (b − c)2 + (c − a)2 + (d − b)2 = (d − a)2.
Find the value of x in each of the following proportions:
51 : 85 : : 57 : x
If 12 L of diesel is consumed by a car in covering a distance of 222 km, how many kilometers will it go in 22 L of diesel?
A gets double of what B gets and B gets double of what C gets. Find A : B and B : C and verify whether the result is in proportion or not
The mean proportional between 4 and 9 is ______.
The table, given below, shows the values of x and y, where x is proportional (directly proportional) to y.
| x | A | 24 | 15 |
| y | 12 | B | 20 |
The values of A and B are:
