SOLUTION: express the foloowing recurring decimals as an infinite G.P. and then find out their values as a rational number. the *x* refer to the recurring part (a)0.*7* (b)0.3*15*

Algebra ->  Sequences-and-series -> SOLUTION: express the foloowing recurring decimals as an infinite G.P. and then find out their values as a rational number. the *x* refer to the recurring part (a)0.*7* (b)0.3*15*       Log On


   

Question 1122513: express the foloowing recurring decimals as an infinite G.P. and then find out their values as a rational number. the *x* refer to the recurring part
(a)0.*7*
(b)0.3*15*
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


(a) x = 0.*7* = 7/10 + 7/100 + 7/1000 + ...

This is a geometric progression with first term 7/10 and common ratio 1/10; the infinite sum is

%287%2F10%29%2F%281-%281%2F10%29%29+=+%287%2F10%29%2F%289%2F10%29+=+7%2F9

Answer: 0.*7* = 7/9

(b) x = 0.3*15* = 3/10 + 15/1000 + 15/100000 + 15/10000000 + ...

After the first term, the rest is a geometric progression with first term 15/1000 and common ratio 1/100; its infinite sum is



Then the entire decimal number is

3%2F10+%2B+15%2F990+=+297%2F990+%2B+15%2F990+=+312%2F990

Answer: 0.3*15* = 312/990; simplify if required.