i am going to show you guys how we can use the simple quadratic equation to find out
the explicit formula for the nth term of the fibonacci sequence so have a look right here
let's consider the quadratic equation and that's the one we have x squared and then minus x minus
two minus one minus three ah minus one uh it's equal to zero hold on let me think yeah i think
this is correct we know this right here is simple but golden because one of the solutions right here
is the golden ratio but for now let me just do the following i'm just going to say let v
you guys know that's the symbol for the golden ratio and we also have another solution
but let's again just look at this i would just say be the solution to this quadratic equation
and because we said that this is the solution for this equation so we are allowed to just put
this back here and we will just end up with v squared and of course it will be minus v minus
1 is equal to 0 but let's put these two terms to the other side so we get v squared equals v plus
one okay pretty good what we want to do next is we'll look at this equation and then multiply
everybody by v that way we can get v to the third power and then v squared plus v so far so good
but hey does this look familiar yes because v squared is just v plus one so we can just go
ahead and put that down which is v plus one and of course we just have to add the v right there
and all together we can say this is just two v plus one okay so far so good i like that
and then let's do the same thing i'm just going to look at this equation here and multiply both
sides by v so that way we can get v to the first power and then we will get v to the third power
plus v squared guess what let's do the same thing because we know v to the third power
which is just equal to that so i can just go ahead and say we will have two v plus one
and then we just have to add v squared and i think we know v square is huh this right here is just
that which is just v plus one and uh can we just go ahead and add them up yeah of course three v
plus two good but did you guys notice what we just did
v to the third power plus v square which is this plus that okay keep that in mind let's do it again
guess what owing all we get five feet plus three good all right let's do it one more time
v to the six part word but this time i'm not going to write down the work let's just look on
the pattern if we can figure this out do you guys see the pattern yes look one
and technically we can also backtrack here is one one two three five so what's the next one
eight yes they are the fibonacci sequence the coefficient of v
from the coefficient the coefficient of v for the fibonacci sequence yes yeah and this right
here should make sense and of course it's valid because the truth is for v6 we're just adding the
previous two expression and that's exactly how we get the fibonacci sequence because for the
next number we add the previous two numbers right so in fact for the next one we just add this that
not together which is five and of course if you want to show all the work these to the sixth power
simply told you so of course this right here will keep on going but now check this out on
the left hand side we are talking about v to a power n so let's just go ahead and write
that down so notice we have v to the nth power this right here will be equal to what
well the coefficients of v right here is the fibonacci sequence and that matches with the n
right because if you have that say six this is actually the sixth fibonacci sequence
one two three four five six so this right here it will be fn
and then we multiply by phi which is the golden ratio and then we add how about the constant terms
where do they come from the truth is they are also the fibonacci sequence but the thing is that they
are just the previous term right the 5 is just the previous term right fine is in front of the h so
for the constant terms it's just f and minus 1 the previous term of the fibonacci sequence
cool now let's come back here again earlier we said let v be the solution and technically we
should say v1 because we have two solutions and the other solution i'm just going to call that
to be v2 so earlier we're just dealing with v1 technically so let me just put down v1 and v1 here
and because v2 is a solution as well so the same thing will happen aha now we have this
kind of system equation to work with and our goal is to find out what fn is check this out
we have f n minus 1 we also have f n minus 1. let's just go ahead and subtract these
two equations i'm just going to do the first one minus the second one and then we will end up with
on the left-hand side we have v1 to the n minus v2 to the n and then on the right-hand side
okay both of them have fn so we can factor that out so i'll just put down f n and then i will just
have v1 minus v2 and then of course this minus that they cancel out nicely cool so now let's
actually figure out what v1 v2s are and of course we just have to solve this quadratic equation and
let's say v1 is the golden ratio which is 1 plus square root of 5 over 2. and then for v2
well this is just 1 minus square root of five over two and right here we will have to get v1 minus v2
this minus that they have the same denominator already so we just do one minus one which is zero
square root of 5 minus a negative square root of 5 is 2 square root of 5
but we have a 2 on the bottom so on all this minus that is just a square root of 5. now ladies and
gentlemen to get this by itself we just have to divide the square root of 5 on both sides
so fn this right here is just going to be 1 over square root of 5 times v1 to the nth power which
is 1 plus square root of 5 over 2 and then to the nth power and then minus v2 which is that 1 minus
square root of 5 over 2 and then to the nth power this right here is indeed the explicit formula
for the nth term of the fibonacci sequence it's super super cool because usually i will show
you guys how to solve the formula by using the difference equation but that's a little
bit technical because you might have to take a discrete map in order to encounter that situation
but for this we saw the pattern earlier and we're able to come with this and then it works out so so
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