26 Sep '11 22:08

The sequence is quite famous, but just in case. It starts with two one's and the rest of the numbers are sums of the two numbers preceding it. That is, the sequence starts with

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

If you divide a number in the sequence with the one that precedes it the ratios converge to the golden ratio, approximately 1.618. Perhaps surprisingly, the same usually applies even if the second number in the series is tweaked. Say, start with 1 and -6 or 1 and 0, you get

1, -6, -5, -11, -16, -27, -43, -70, ...

1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

again, the ratios converge to the golden ratio.

The question is.. if you tweak the 2nd number in the Fibonacci sequence, getting

1, x, x+1, 2x + 1, 3x + 2, 5x + 3 ...

...are there any values for x for which the sequences does NOT converge to the golden ratio? If not, why not; if yes, find such a value.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

If you divide a number in the sequence with the one that precedes it the ratios converge to the golden ratio, approximately 1.618. Perhaps surprisingly, the same usually applies even if the second number in the series is tweaked. Say, start with 1 and -6 or 1 and 0, you get

1, -6, -5, -11, -16, -27, -43, -70, ...

1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

again, the ratios converge to the golden ratio.

The question is.. if you tweak the 2nd number in the Fibonacci sequence, getting

1, x, x+1, 2x + 1, 3x + 2, 5x + 3 ...

...are there any values for x for which the sequences does NOT converge to the golden ratio? If not, why not; if yes, find such a value.