Q8 of 176 Page 20

Find the rational numbers having the following decimal expansions :

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


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