## Friday, August 11, 2006

### This is tricky

Add up these ten numbers without a calculator:
6
12
18
30
48
78
126
204
330
534

Spoilers below!!!

Hint: there's a trick to this...

Hint3: You can answer this question using the seventh number in the series.

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1. 1386 - it was easier to add the numbers in my head than try to figure out a trick to this.

2. The trick is to multiply the seventh number by 11.

The sequence is a sort of fibonacci sequence: each number is the sum of the two previous (not the classic version starting with 1,1,2,3,5,8...). If you represent such a sequence algebraically with, say, the first two numbers as "a" and "b", forming each subsequent member by adding up the two previous, you get this:

a
b
a+b
a+2b
2a+3b
3a+5b
5a+8b
8a+13b
13a+21b
21a+34b

If you add up all of these, you get 55a+88b, which is obviously the seventh number (5a+8b) times 11. So, in any fibonacci-like sequence consisting of ten numbers, the sum can always be found by multiplying the seventh number times 11, which, in this case, is 126*11=1386.

3. Part of my summer reading was Mario Livio's "The Golden Ratio." He has a nice little riff on this oddity in Chapter 5, page 104 in the 2002 paperback edition. It's good book, and worth a look if you haven't seen it.

4. Abe explained it better than I ever could have.

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