+0 The base-three repeating decimal 0.12 is equivalent to what base-ten common fraction?
+129830 We can write this as the sum of two series
First series = 1*3^(-1) + 1*3^(-3) + 1*3^(-5) + ......
The sum of this series is [ 1*3^(-1) ] / [ 1 - 3^(-2) ] = (1/3) / [ 8/9] = 9/24 = 3/8
Second series = 2*3^(-2) + 2*3^(-4) + 2*3^(-6) + ......
The sum of this series is [ 2*3^(-2) ] / [ 1 - 3^(-2)] = (2/9) / (8/9) = 1/4
The sum is (3/8) + (1/4) = [12 + 8 ] / 32 = 20 /32 = 5/8 = . 12 (repeating) base 3
+129830 We can write this as the sum of two series
First series = 1*3^(-1) + 1*3^(-3) + 1*3^(-5) + ......
The sum of this series is [ 1*3^(-1) ] / [ 1 - 3^(-2) ] = (1/3) / [ 8/9] = 9/24 = 3/8
Second series = 2*3^(-2) + 2*3^(-4) + 2*3^(-6) + ......
The sum of this series is [ 2*3^(-2) ] / [ 1 - 3^(-2)] = (2/9) / (8/9) = 1/4
The sum is (3/8) + (1/4) = [12 + 8 ] / 32 = 20 /32 = 5/8 = . 12 (repeating) base 3
