Find the rational numbers having the following decimal expansions :

Let,

x = 3.522222222 …..

x = 3.5+0.02 + 0.002 + 0.0002 + …∞

⇒ x = 3.5+2(0.01 + 0.001 + 0.0001 + …∞ )

⇒ x =

⇒ x = 3.5 + 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 the sum of infinite GP.

⇒ S =

∴ x = 3.5 + 2(1/90)

⇒ x = (35/10) + 1/45 = (315+2)/90 = 317/90