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