Why does this work

Adding a series of ten numbers
1. Have a friend choose and write down a single-digit number. (Two digits for experts.)
2. Ask your friend to name and note a third number by adding the first two.
3. Name a fourth by adding the second and third. Continue in this way, announcing each number, through ten numbers.
4. Ask your friend to add up the ten numbers. You will give the answer before he or she can finish:
The sum of all the terms of this series will be the seventh number multiplied by 11.
Example:
1. If the numbers selected are 7 and 4:
2. The series jotted down is: 4, 7, 11, 18, 29, 47, 76, 123, 199, 322.
3. The seventh number is 76. 11 x 76 = 836
(use the shortcut for 11: 7 is the first digit, 6 is the third digit;
the middle digit will be 7 + 6, and carry the 1: 836).
4. So the sum of the ten numbers is 836.

we would like to know why this works explaining with a formula. could you please help us! thanks

two desperate students!!!!

John a
Leerling bovenbouw havo-vwo - donderdag 4 december 2003

Antwoord

So we are going international!
Very well then.
First, there's a line missing, because (according to the example) you need TWO numbers to start with.
The formula is not so difficult.
Say the first two numbers are a and b.
Then the ten numbers will be:
a
b
a+b
a+2b
2a+3b
3a+5b
5a+8b
8a+13b
13a+21b
21a+34b
(by the way: do you recognise Fibonacci's numbers?)
Just add up these numbers, and verify that this sum equals 11 times the seventh number.
It doesn't matter how many digits you choose for a and b, but the multiplication by 11 is a little tougher for larger numbers.
Good luck, and don't despair.

Zie Adding a series of ten numbers

donderdag 4 december 2003

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