Fibonacci numbers - Math Central

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Question from John:

Make a single column of numbers. Start with two numbers of your choice.
The third number is the sum of the previous two, the fourth number is the sum of
numbers two and three, and so on until
you have ten numbers in the column. Add up all ten numbers. Now, take the
seventh number and multiply it by eleven. This product will equal the sum of the
ten numbers. The same result will occur regardless of the first two numbers
chosen.
The question is why does the 7th number multiplied by 11 always equal the sum of
the ten numbers?

Hi John,

Did you try it?

The Fibonacci Sequence starts with 1 and 1 so the first ten terms are

1 1 2 3 5 8 13 21 34 55

Adding these I get 143. The seventh term is 13 and $13 \times 11 = 143$ so it works for the Fibonacci Sequence.

What if you start with -1 and 3 then the sequence starts

-1

-1 + 3 = 2

3 + 2 = 5

2 + 5 = 7

5 + 7 = 12

7 + 12 = 19

and so on. The seventh term is 19 and $19 \times 11 = 209.$ If you continue the sequence to 10 terms do you get a sum of 209?

What if you don't know the first two numbers in the sequence? The standard thing to do in algebra is to let two letters represent the two unknown numbers.

Let the first two numbers in the sequence be $k$ and $n.$ then the first six terms on the sequence are

$k$

$n$

$k + n$

$n + k + n = k + 2n$

$k+n + k + 2n = 2k + 3n$

$k + 2n + 2k + 3n = 3k + 5n$

and so on.

What is the seventh term? What is 11 times the seventh term?

Continue the sequence to ten terms. What is the sum of the ten terms?

Write back if you need more assistance,
Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.