Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Find the fourth proportion to the following :
(x2 - y2),(x3 + y3)anc(x3 - xy2 + x2y- y3)
If a, b, c, d are in continued proportion, prove that:
(a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2.
If `x/a = y/b = z/c`, show that `x^3/a^3 - y^3/b^3 = z^3/c^3 = (xyz)/(zbc).`
Verify the following:
39 : 65 : : 141 : 235
35 inland letters cost Rs 87.50. How many such letters can we buy for 315?
If a + c = mb and `1/b + 1/d = m/c`, prove that a, b, c and d are in proportion.
If a, b, c and d are in proportion, prove that: `(a^2 + ab)/(c^2 + cd) = (b^2 - 2ab)/(d^2 - 2cd)`
If a, b, c are in continued proportion, prove that: a : c = (a2 + b2) : (b2 + c2)
If a, b, c, d are in continued proportion, prove that: `((a - b)/c + (a - c)/b)^2 - ((d - b)/c + (d - c)/b)^2 = (a - d)^2 (1/c^2 - 1/b^2)`.
Which of the following statements correctly defines a proportion involving four non-zero quantities a, b, c, and d?
