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Number Properties: Help & Review8 chapters | 55 lessons

Instructor:
*Gerald Lemay*

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, individual Fibonacci numbers are related by the Cassini identity. We clarify what this identity means and show how to prove it using the method of induction.

In normal everyday language, ''identity'' is the who, what, where, when and why about a person. For example, Cassini's identity: Giovanni Domenico Cassini, an Italian astronomer and mathematician who lived from 1625 to 1712, famous for the discovery of the ring divisions of the planet Saturn and for the math identity which bears his name.

The topic for this lesson is a math identity. In math, an **identity** states something is equal to something else. It's okay to think of a math identity as an equation with one or more variables.

In this lesson we explore Cassini's (math) identity.

Before we can understand and prove Cassini's identity, we need to review Fibonacci numbers. By the way, the Italian mathematician, Leonardo Fibonacci, lived 4 centuries before Cassini.

Take the numbers 0 and 1. Add 1 to 0 and we get 1. Ignoring the zero, we have the sequence 1, 1.

Add the last number (1) to the previous number (1) and we get 1 + 1 = 2. Now the sequence is 1, 1, 2.

Add the last number (2) to the previous number (1) and we get 2 + 1 = 3. The sequence is 1, 1, 2, 3.

If we continue this process, the sequence becomes 1, 1, 2, 3, 5, 8, 13, 21, … These are the **Fibonacci numbers**.

It's nice to keep track of where we are in the sequence. We can use intergers 0, 1, 2, 3, …

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |

F |
1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 |

*F* is the Fibonacci number. Thus, *F*1 is 1 and *F*4 is 5. What is *F*6? Right, 13.

Cassini brilliantly observed:

This is the **Cassini identity**.

What is this saying? First of all, the variable in this identity is *n* and the values it can have are integers starting at 1. Thus, *n* can be 1, 2, 3, …

Looking at the right-hand side of Cassini's identity, what happens when we raise (-1) to the *n* + 1 power? It all depends on the value of *n*. For example, if *n* is an odd number, like 1, 3, 5, … then (-1)*n* + 1 = +1. You can check with *n* = 1 which gives (-1)1 + 1 = (-1)2 = +1. How about even numbers for *n* like 0, 2, 4, …. Right! We get -1.

Delving a little deeper, let's verify for a particular value of *n* ≥ 1. How about *n* = 1? Okay, for *n* = 1,

*F**n*+ 1 is*F*1 + 1 which is*F*2 which is 2 (from the table)*F**n*- 1 is*F*1 - 1 is*F*0 which is 1*F**n*is*F*1 which is 1

Substituting into the left-hand side of Cassini's identity:

How about the right-hand side?

- (-1)
*n*+ 1 is (-1)1 + 1 (-1)2 which is +1

Substituting:

Left-hand side equals the right-hand side. Thus, we have verified the identity for *n* = 1.

Just for fun, take a moment to verify the identity for *n* = 6.

Here are the details. For the left-hand side:

and for the right-hand side:

Again, left-hand side equals the right-hand side!

Instead of verifying for all the other values of *n* (which would take forever), we use a really nice proof method.

A **proof by induction** has the following steps:

1. verify the identity for *n* = 1

2. assume the identity is true for *n* = *k*

3. use the assumption and verify the identity for *n* = *k* + 1

4. explain and make a conclusion:

- if we backtracked to
*n*= 1 (verified in step 1), this*n*= 1 could be our*k* - the identity was verified for
*n*=*k*+ 1 provided*n*=*k*was true. Therefore, the identity is verified for all integer values of*n*≥ 1. And the proof is done!

Not so bad! We've already done step 1 verifying for *n* = 1. On to step 2.

Replace *n* with *k*:

and assume this to be true.

In step 3, we go to the next value of *n*. After *n* = *k* we have *n* = *k* + 1.

Replacing *n* with *k* + 1:

and simplifying:

The next portion of step 3 is to figure out a way to use the assumption.

We could start by replacing *F**k* + 2.

On the left-hand side:

We used Fibonacci where *F**k* + 2 is the sum of the two previous numbers, *F**k* + 1 and *F**k*.

Continuing with the left-hand side, we expand:

Almost there! The last term, (*F**k + 1* )2 becomes

*F**k*+ 1*F**k*+ 1 which is*F**k*+ 1 (*F**k*+*F**k*- 1 ) using Fibonacci. Then, expanding*F**k*+ 1*F**k*+*F**k*+ 1*F**k*- 1

Let's review. From the left-hand side we have:

which becomes

and then

leaving

which can be re-ordered as:

This really looks like our assumption! We can replace *F**k* + 1 *F**k* - 1 with (-1)*k* + 1. Thus,

Writing (-1)1 as the negative in front of the parentheses:

Then, (-1)1 (-1)*k* + 1 = (-1)*k* + 2.

We have started with the left-hand side for *n* = *k* + 1 and used the assumption and Fibonacci. The result is identical to the right-hand side for *n* = *k* + 1. Okay, step 3 is finished. Thus, if *n* = *k* is true, then so is *n* = *k* + 1.

In the last step, we restate *n* = *k* + 1 is true provided the *n* = *k* case is true. Since, *k* is arbitrary, it could have been *n* = *k* = 1. This *n* = 1 case was verified. Thus, the next value for *n*, *n* = 2 is true. So is *n* = 3, …

This concludes the proof by induction of Cassini's identity.

An **identity** is a mathematical equation with one or more variables. The **Cassini identity** has the variable *n*:

where *F* are the **Fibonacci numbers**. In a Fibonacci sequence, the next number is the sum of the current number and the previous number. Rather than verifying an identity for all possible values of *n*, we use a **proof by induction**. In this method, the *n* = 1 case is verified. Then, we assume the *n* = *k* case is true. Accepting the assumption as true, we show the *n* = *k* + 1 case is also true. Since *n* = 1 was shown to be true and since we have shown the next *n* is true if the current *n* is true, then *n* = 2 is also true. And so are the cases for *n* = 3, 4, 5, … which inductively proves the identity.

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Number Properties: Help & Review8 chapters | 55 lessons

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