Find the rational numbers having the following decimal expansions :
Let,
x = 0.33333333…..
x = 0.3 + 0.03 + 0.003 + …∞
⇒ x = 3(0.1 + 0.01 + 0.001 + …∞ )
⇒ x =
We observe that the above progression possess a common ratio. So it is a geometric progression.
Common ratio = 1/10 and first term (a) = 1/10
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.
⇒ x = 3×
∴ x = 1/3