Fnk Fnk x

Hint: consider even values of k then look at the odd values of k.

8. Surprisingly, there is a similar formula for the Lucas numbers L(n-k)+L(n+k).

Repeat the above investigation for this new expression, spotting the patterns for k=1, then k=2, k=3, k=4, and so on, until you can spot the general pattern.

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 ..More..

A number trick based on Phi, Lucas and Fibonacci numbers!

Here is a trick that you can use to amaze your friends with your (supposed) stupendous calculating powers. All you need to remember is a few Lucas and Fibonacci numbers and you can write down a complicated expression like this:

You can ask them to verify these formulas on their calculators and they will always work out! The 4 by the y sign means the fourth-root. So if

24 =16 "2 to the fourth is 16" then 2 = Vl6 ' 2 is the fourth-root of 16"

You will often find a button on your calculator which extracts roots (perhaps marked Vx) near the button which computes the power of a number (marked xy). If there is no ^/x button on your calculator, you can compute for instance by computing 1/4 first and using this as the y power with x as 16. This is because yyx =

What's the secret?

You will need to learn a few of the early Lucas and Fibonacci numbers and their positions in the sequences:

n:

0 1

2 3

4

5

6 7 8 9 10 ...

Fn:

0 1

1 2

3

5

8 13 21 34 55 ...

Ln:

2 1

3 4

7

11

18 29 47 76 123 ...

For the example at the head of this section, I randomly picked the index (column) 4 numbers, F(4)=3 and L(4)=7. We will use these three numbers, 4, 3 and 7 in both expressions. Notice that the first expression has a plus inside its 4-th-root-sign whereas the second has a minus.

Since the position number, 4, is EVEN, I will use a MINUS sign BETWEEN the two expressions.

Now just substitute your values into this formula:

The SIGN in the middle is + if n is ODD and - if n is EVEN

Here is a Fibonacci and Lucas Numbers Calculator which also generates these expressions for you. Click on the "Amaze me!" button and see a new example every time.

An even more complicated-looking variation!

If you want to make it look even more complicated, choose TWO columns in the table, one for the first expression and one for the second. Here's an example where I use the fifth and ninth columns:

The sign in the middle (between the two root-expressions) will depend on the SECOND POSITION (in the example it was 9): if it is ODD (and 9 here is odd), then use PLUS and if it is EVEN put a MINUS sign.

In the new example above, I chose two different positions: 5 for the first expression and 9 for the second.

For the first expression with position=5, I will then use Fib(5)=5 and Lucas(5)=11.

For the second, with position 9, I will use Fib(9)=34 and Lucas(9)=76.

Since 9, the second choice, is ODD, I will put a PLUS sign between the two expressions.

Just substitute your two sets of values: N, Lucas(N) and Fib(N); K, Lucas(K) and Fib(K) in each expression like this, taking care not to mix up your two sets of numbers:

REMEMBER that the first expression always has a plus(+) inside the root sign and the second always has a minus (-) inside its root-sign but the sign in-between depends on the second (K) value.

Why does it work?

Follow through the suggestions in the following Investigation section and the secret will be revealed!

1. (a) See what happens in the first n-th-root expression if we let n=2. The first expression is just:

Use your calculator and find its value.

(b) Now try the second expression with n (or k) =2:

Use your calculator and find this value.

(c) Adding the numbers in (a) and (b) should give 1. Does it?

2. Repeat the above for n=3 finding the two values:

Check that combining them really does give 1, remembering that since n is ODD, you must subtractthe second from the first, not add it.

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