# Need help figure out a Fibonacci related math trick

My math teacher used to do a trick where he would have a student write $$2$$ numbers on the board then add the first to the second to create the third then add the second to the third and so on until there were $$10$$ numbers. He would then turn around and add them up in $$2$$ seconds. How did he do this?

• Have you tried a few simple cases to see the pattern? – rtybase Sep 5 '19 at 20:04
• have you heard of induction? – ggg Sep 5 '19 at 20:06
• Check my answer, I think it is easy; the seventh number in the list multiplied by 11. – Hussain-Alqatari Sep 5 '19 at 21:55

Multiplying any natural number by $$11$$ is so easy, check here.

Now the solution for your problem is to multiply the $$7^\text{th}$$ number in the list by $$11$$

Have this example: our first two numbers are $$16$$ and $$21$$

So the list is:

$$16$$

$$21$$

$$37$$

$$58$$

$$95$$

$$153$$

$$248$$

$$401$$

$$649$$

$$1050$$

The sum of those numbers is just $$248$$ (which is the $$7^\text{th}$$ number) $$\times 11=2728$$.

The rule is: $$\boxed{7^\text{th}\text{ number }\times 11}$$

Try it algebraically starting with $$a$$ and $$b$$ $$\begin{eqnarray*} a,b,a+b,a+2b,2a+3b,3a+5b,5a+8b, \\ 8a+13b,13a+21b,21a+34b. \end{eqnarray*}$$ Now add these together and we get $$55a+88b=11 (5a+8b)$$.

So I guess your teacher took the first value multiplied by $$5$$ and added it to the second value multiplied by $$8$$ and then multiplied by $$11$$. Your teacher would have had plenty of time to do this calculation while then values were being added.

• There's no need for the teacher to calculate $5a+8b$ - it's already written on the board! – Carmeister Sep 6 '19 at 4:27
• Actually the teacher can calculate $5a+8b$ essentially faster than it will be written on the board, and have additional time to perform multiplication by $11$ (and check the answer twice). – Oleg567 Sep 7 '19 at 9:14

Hint:

$$\begin{array}{rl} F(1) &= \color{blue}{F(3)}-F(2)\\ F(2)&= F(4)\color{blue}{-F(3)}\\ F(3)&=\color{red}{F(5)}-F(4)\\F(4)&=F(6)\color{red}{-F(5)}\\ \vdots\end{array}$$

$$F(1)+F(2)+\dots+F(n) = F(n+2)-F(2)$$

That is because Fibonacci numbers have a number of properties, one of them being:

$$\sum_{i=0}^nF_i = F_{n+2} - 1 = 2F_n + F_{n-1} - 1$$

Proof is by induction

Hence, if the numbers are $$0,1,1,2,3,5,8,13$$, the sum will be $$13*2 + 8 - 1 = 33$$

Well, to answer the question as to how he did it: If the first number is $$x$$ and the second number is $$y$$ then every other number and the sum of all ten numbers will a combination of $$x$$ and $$y$$. As you do the same thing every time the final sum will be the same combination. Your teacher merely memorized that the final sum would be $$55x + 88y$$.

As to how we would know the final number is $$55x+88y$$ we can

1) Simply do it. The ten numbers are $$x,y,x+y, x+2y, 2x+3y, 3x+5y,5x+8y,8x+13y,13x+21y, 21x+34y$$ and the sum is $$55x+88y$$.

2) Try to find a way to generalize this without doing each sum.

We notice the number of $$x$$s involved are $$1,0,1,1, 2,etc.$$. After a slow start once we have $$1,1$$ this has to follow the Fibonacci sequence. So if the $$k$$th number is $$a_k x + b_k y$$ we know $$a_k= F_{k-2}$$, the $$k$$th fibonacci number.

We notice the number of $$y$$s involved are $$0,1,1,2, etc.$$ and those are the Fibonacci numbers too but with a quicker start. $$b_k = F_{k-1}$$.

So the $$k$$th number is $$F_{k-2}x + F_{k-1}y$$.

The final total after ten numbers is therefore $$(1 +\sum_{k=1}^8 F_k)x + (\sum_{k=1}^9 F_k) y$$.

There's an interesting formula that

$$\sum_{k=1}^n F_k = F_{n+2} - 1$$.

So the sum is $$F_{10}x + (F_{11}-1)y$$

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Another answer states that the answer is that the sum is the $$7$$th number times $$11$$.

So does $$11(F_{5}x + F_{6}y) = F_{10}x + (F_{11} -1)y$$?

Well, $$11(F_5x + F_6y) = 11(5x + 8y)=55x + 88y = 55x + (89-1)y$$.

The Fibonacci series is $$1,1,2,3,5,8,13,21,34,55,89$$ so, yes, indeed this is true.

So that's actually how the teacher did it so quickly. You wrote down the $$7$$th term and he multiplied it by $$11$$ in his head.

• Thanks everyone! – PotatoHeadz35 Sep 6 '19 at 11:45