The ratio of the A.M. and G.M. of two positive numbers a and b is m : n. Show that

a : b = (m + √(m² − n²)) : (m − √(m² − n²))

Let Sₙ = 0.6 + 0.66 + 0.666 + … to n terms

= 6[0.1 + 0.11 + 0.111 + … to n terms]

= (6/9)[0.9 + 0.99 + 0.999 + … to n terms]

= (6/9)[(1 − 1/10) + (1 − 1/10²) + (1 − 1/10³) + … to n terms]

= (2/3)[(1 + 1 + … n terms) − (1/10)(1 + 1/10 + 1/10² + … n terms)]

= (2/3)[n − (1/10)((1 − (1/10)ⁿ) / (1 − 1/10))]

= (2/3)n − (2/30) × (10/9)(1 − 10⁻ⁿ)

= (2/3)n − (2/27)(1 − 10⁻ⁿ)