Question

Question 29 If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are A ± [ (A+G)(A-G) ]^(1/2).

Class X1 - Maths -Sequences and Series Page 193

Answer 1

Let the numbers are α and β
Then,
(α + β)/2 = A
(α + β) = 2A

and,
$\sqrt{\alpha.\beta}=G\\\alpha.\beta=G^2$

Now, quadratic equation having roots α and β is
x² - (α + β)x + α.β = 0
x² - 2A + G² = 0 { after putting above value }
now, use formula for finding roots of quadratic equation.
$x=\frac{2A\pm\sqrt{4A^2-4.1.G^2}}{2.1}\\=\frac{2A\pm2\sqrt{A^2-G^2}}{2}\\=A\pm\sqrt{(A+G)(A-G)}$

Answer 2

your answer .........