Math Is Fun Forum

I think your reasoning is perfectly sound, and deducible from the rules given that
1. The number of even terms is the nth Fibonacci number
2. The number of odd terms is the n-1 Fibonacci number, and
3. The total number of terms is the n+1 Fibonacci number
For any given Sn.

Just to confirm, I will call a Fibonacci number that corresponds to the step number Fn, and restate exactly what you said in those terms.
Sn-1 has Fn-2 odds and Fn-1 evens.
Then Sn has Fn-1 odds and Fn = Fn-2 + Fn-1 evens,
The total number of elements being Fn+1 = Fn-2 + 2Fn-1
Sn+1 has Fn = Fn-2 + Fn-1 odds and Fn+1 = Fn-2 + 2Fn-1 evens

To slightly correct your running counts, it goes:
Odds: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34...
Evens: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...
Total: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...

As far as I'm concerned, the only outstanding problem is how to obtain a general formula for the sum (there is also a little problem of obtaining a general formula for the smallest number, but that should be pretty easy). Everything else is toast by now, and it all has to do with Fibonacci.

A final note: A more intuitive way of solving 3. than just using the formula is to do what the "Prince of Mathematicians", Carl Gauss, apparently did as a schoolboy. He was told to write out the sum of natural numbers up to 100, and quickly solved it by recognising that most of the numbers could be matched up in pairs that add up to 100, for example 100+0=100, 99+1=100, 98+2=100... right down to 51+49 = 100, counting fifty pairs plus 50, giving a speedy answer of 5050.

The same can be done with sums up to 25, or any amount. With 25+0=25, 24+1=25, 23+2=25... you will promptly get to 13+12=25 and count thirteen pairs.

The sequence in 1. is not arithmetic or geometric; there is no common difference or common ratio. The pattern for 1. seems to be 1 + 2 + 3 + 4... These numbers are called triangular numbers, and I think the Pythagoreans revered them over 2500 years ago. There is a polynomial rule for this, it is

For 2., you are correct.

Hey! I am very ignorant of geography even with so many clues, but I finally know one xD

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Hi;

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Hey!

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Oh, I see Thank you, I was just completely unfamiliar with this type of problem and it took a while to see what was going on.

Oh... I knew there must be some hocus-pocus I was not privy to. Thank you for mentioning that