Let,

x = 0.688888888888…..

x = 0.6+0.08 + 0.008 + 0.0008 + …∞

⇒ x = 0.6+8(0.01 + 0.001 + 0.0001 + …∞ )

⇒ x =

⇒ x = 0.6 + 2S

Where S =

We observe that the above progression possess a common ratio. So it is a geometric progression.

Common ratio = 1/10 and first term (a) = 1/100

Sum of infinite GP = ,where a is the first term and k is the common ratio.

Note: We can only use the above formula if |k|<1

∴ we can use the formula for sum of infinite GP.

⇒ S =

∴ x = 0.6 + 8(1/90)

⇒ x = (6/10) + 4/45 = (54+8)/90 = 62/90