Please please please help me please

Each term is a sum of the two previous terms like the fibonacci sequence, but the sum of the two starting terms of the lucas sequence is followed by 2 then 1, rather than 0 then 1 for the fibannaci sequence.

These are integer sequences.

The sum of the two previous terms can be represented by:

fn = fn-1 + fn-2.

These numbers can be obtained by adding the previous corresponding term of the fibonacci sequence by the next corresponding term to get the corresponding number of the lucas sequence.

Ln = { 2 if n = 0

1 if n = 1

Ln-1 + Ln-2 if n > 1

Fn+1 + Fn-1 = Ln

f(0) = -1

f(1) = 2, f(2) = 1, f(3) = 3, f(4) = 4, f(5) = 7, f(6) = 11

f(12) = 199

f(12) = f(11) + f(10) = (f(10) + f(9)) + (f(9) + f(8)) =

given f(n) = f(n-1) + f(n-2)

f(12) = f(1) + f(2) = f(2) + f(3) = f(3) + f(4) =

f(4) + f(5) = f(5) + f(6) = f(6) + f(7) =

f(7) + f(8) = f(8) + f(9) = f(9) + f(10) =

f(10) + f(11) = f(12) →

f(12) = 2 + 1 = 1 + [3] = 3 + [4] = 4 + [7] =

7 + [11] = 11 + [18] = 18 + [29] = 29 + [47] =

47 + [76] = 76 + [123] = 199

Explicit formula

for n is an integer.

Nth term of the lucas sequence.

As you can see, the golden ratio

( 1 + √5 / 2) appears in here.

L(n) = ((1 + √5) / 2)ⁿ⁻¹ + ((1 – √5) / 2)ⁿ⁻¹

Hello, n being an integer, we need to prove that one statement depending on n is true, let's note it S(n).

The mathematical induction involves two steps:

Step 1 - We need to prove S(1), meaning that the statement is true for n = 1

Step 2 - for k integer > 1, we assume S(k) and we need to prove that S(k+1) is true.

Imagine that you are a painter and you need to paint all the trees on one side of a road. You have several colours that you can use but you are asked to follow two rules:

Rule 1 - You need to paint the first tree in white.

Rule 2 - If one tree is white you have to paint the next one in white too.

What colour do you think all the trees will be painted?

Do you see why this is very important to prove the two steps as well ?

Let's do it in this example.

Step 1 - for n = 1, let's prove that S(1) is true, meaning  

So the statement is true for n = 1

Step 2 - Let's assume that this is true for k, and we have to prove that this is true for k+1

So we assume S(k), meaning that

and what about S(k+1), meaning ?

We will use the fact that this is true for k,

We can write it because the statement at k is true and then we can conclude.

In conclusion, we have just proved that S(n) is true for any n integer greater or equal to 1, meaning

Hope this helps.

Do not hesitate if you need further explanation.

Thank you