Amazing or expected?
1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321
1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 +10= 1111111111
9 x 9 + 7 = 88
98 x 9 + 6 = 888
987 x 9 + 5 = 8888
9876 x 9 + 4 = 88888
98765 x 9 + 3 = 888888
987654 x 9 + 2 = 8888888
9876543 x 9 + 1 = 88888888
98765432 x 9 + 0 = 888888888
Amazing or expected?
Source: Desi Baba’s Get The Knowledge Spread The Knowledge.
November 28th, 2008 at 2:43 am
1/(1-x) = 1 + x + x² + x³ +…
1/(1-x)² = (1 + x + x² + x³ +…)(1 + x + x² + x³ +…)
= 1 + 2x + 3x² + 4x³ +…
1/(1/x – 1) = x + x² + x³ + x^4 + …
1/(1/x – 1)² = x² + 2x³ + 3x^4 + 4x^5 + …
Put x=1/10, so 1/x=10. Then decimal notation reveals the coefficients so long as there are no carries between one place and the next.
1/x – 1 = 9.
1/9 = .11111111…. recurring
1/81 = .012345679 recurring.
1/9 – 1/81 = (9-1)/81 = 8/81 = .098765432 recurring.
Now here’s a more complicated example. I’ve in effect used base 100 to reveal more terms of the sequences.
1/9899 = .00 01 01 02 03 05 08 13 21 34 55…. (Fibonacci numbers)
102/9899 = .01 03 04 07 11 18 29 47…. (Lucas numbers)