Recurrence relations and sequences

[naav]

Hi...

1. so can i say that a recurrence relation is a description of the operation(s) involved in a sequence...?...

2. is the formula for an arithmetic sequence, a recurrence relation...?...

and is the formula for a geometric sequence,
a recurrence relation...?...

 

[tomkeus]

1.Say rather that it is a rule which forms sequence.
2.If you mean $$a_{n+1}=a_n+d$$, then it is.
3.If you mean $$a_{n+1}=qa_n$$, then it is.

Any form like this:
$$a_1=x$$
$$a_{n+1}=f(a_n...a_1)$$
is a recurrence relation.

 

[matt grime]

It needn't be just a function of the previous term in the sequence. that is a first order recurrence relation, for want of a better term (think first order differential equation). things such as the fibonacci numbers satisfy a degree two difference equation (recurrence relation):

a_1=1, a_2=1, a_n=a_{n-1}=a_{n-2} for n>2,

for example

 

[naav]

Hi...thanks...

i was told that a recurrence relation expresses a term in the sequence with regards to other terms in the sequence...whereas the formulae for the arithmetic and geometric sequences don't...

...?...

 

[Gokul43201]

Hi...thanks...

i was told that a recurrence relation expresses a term in the sequence with regards to other terms in the sequence...whereas the formulae for the arithmetic and geometric sequences don't...

...?...

Yes, for an arithmetic sequence, you would say that

$$a_n = a_{n-1} + d,~~ n = 1,2,3,...$$

This is how an AS is defined. But it's not hard to figure from here, that

$$a_n = a_1 + (n-1)d , ~~for~~ n=1,2,3,...$$

 

[naav]

Hi...thanks...

1. so i guess there's a similar recurrence relation for a geometric sequence...?...

2. so then why have a recurrence relation when you can express the SAME sequence by a formula...?...

 

[Gokul43201]

Hi...thanks...

1. so i guess there's a similar recurrence relation for a geometric sequence...?...

2. so then why have a recurrence relation when you can express the SAME sequence by a formula...?...

Yes, for a Geometric Sequence,

$$a_n=r*a_{n-1}, ~n=1,2,3,...$$
$$~~~ = a_1*r^{n-1}$$

Sometimes, it hard (or impossible ) to find a formula for the n'th term, but you can describe the entire series by a recurrence relation. For the Fibonacci Sequence (decribed by matt, above) it's hard to find such a formula (though there is a good approximation that works well for the large terms ).

 

[e(ho0n3]

For the Fibonacci Sequence (decribed by matt, above) it's hard to find such a formula (though there is a good approximation that works well for the large terms ).

I'm going to disagree here. The Fibonacci recurrence is a simple second-order linear homogenous recurrence relation with constant coefficients (both of them being 1). These types of recurrences are easily 'solvable'. The explicit formula for the Fibonacci sequence involves the Golden Ration by the way, which I find extremely curious.