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प्रश्न
If A is the arithmetic mean and G1, G2 be two geometric means between any two numbers, then prove that 2A = `(G_1^2)/(G_2) + (G_2^2)/(G_1)`
उत्तर
Let the two numbers be x and y
∴ A = `(x + y)/2` ......(i)
If G1 and G2 be the geometric means between x and y then x, G1, G2, y are in G.P.
Then y = xr4-1 ....[∵ an = arn–1]
⇒ y = xr3
⇒ `y/x` = r3
⇒ r = `(y/x)^(1/3)`
Now G1 = xr
= `x(y/x)^(1/3)` .....`[because r = (y/x)^(1/3)]`
And G2 = xr2
= `x(y/x)^(2/3)`
∴ From R.H.S. `G_1^2/G_2 + G_2^2/G_1 = (x^2(y/x)^(2/3))/(x(y/x)^(2/3)) + (x^2(y/x)^(4/3))/(x(y/x)^(1/3))`
= `x + x(y/x)^(4/3 - 1/3)`
= `x + x(y/x)`
= x + y = 2A L.H.S. ....[Using equation (i)]
∴ L.H.S. = R.H.S.
Hence proved.
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