Sum of 10 numbers

[Mark Kudlowski]

Showed this maths trick to Rachel when I was in the Countdown audience recently :

RR to players:

Start with any two numbers, say 7 and 1, and write them in a column.
Add them to get 8, and put 8 in the third column.
Add 1 and 8 to get 9, put in 4th column.
8 + 9 = 17, so put 17 in the 5th column.
9 + 17 = 26, so put 26 into the 6th column.
Repeat this same process until you have 10 numbers in the column; 7, 1, 8, 9, 17, 26, 43, 69, 112, 181. (An example of a generalised "Fibonacci" sequence)

But .... before the 10th number has been entered, RR writes 473 on a card, lets the players sum the 10 numbers together, and
those selfsame 10 numbers add to 473 !

Start with any other two numbers, say 1 and 1 (THE Fibonacci sequence) :

1, 1, 2, 3, 5, 8, 13, 21 ..... - I can say the total of the first 10 numbers is 143 without writing the whole list.

To the mathematicians among you, how and why does this trick work ?

[Sean Fletcher]

I've spotted a link between the 7th number of the sequence being multiplied by 11 to give the sum of the first ten numbers. That is the how. I'm stuck on the why though!

[Mark Kudlowski]

I've spotted a link between the 7th number of the sequence being multiplied by 11 to give the sum of the first ten numbers. That is the how. I'm stuck on the why though!

The proof is on this document : (Paste the URL)

http://mkhometuition.co.uk/index_files/ ... _Proof.pdf

Top left of the front page graphic
Full details on Example(2) on Page 3

[Sean Fletcher]

That seems really obvious now I know the why. Cool trick!