Recursive definitions are a cool way to describe sequences. They let us create new terms using the ones that came before. This method helps us get a better grip on different kinds of sequences, especially when you're in Grade 12 Pre-Calculus.
Arithmetic Sequences:
This type uses the rule:
( a_n = a_{n-1} + d )
Here, ( d ) is the common difference.
For example, if we start with ( a_1 = 2 ) and say ( d = 3 ):
The sequence goes like this: ( 2, 5, 8, 11, \ldots )
You can find the total number of terms with this formula:
( a_n = a_1 + (n-1) \cdot d )
Geometric Sequences:
These follow a different rule:
( a_n = a_{n-1} \cdot r )
Here, ( r ) is the common ratio.
For example, if ( a_1 = 3 ) and ( r = 2 ):
The sequence looks like this: ( 3, 6, 12, 24, \ldots )
To find any term in this kind of sequence, you can use:
( a_n = a_1 \cdot r^{n-1} )
Fibonacci Sequences:
In this case, we use the rule:
( F_n = F_{n-1} + F_{n-2} )
with the first two terms as ( F_1 = 1 ) and ( F_2 = 1 ).
This creates the sequence:
( 1, 1, 2, 3, 5, 8, 13, \ldots )
Harmonic Sequences:
These are made from the inverses (or “flips”) of an arithmetic sequence.
For example, starting with ( 1, 2, 3, \ldots ) gives us the harmonic sequence:
( { 1, \frac{1}{2}, \frac{1}{3}, \ldots } )
Using recursive definitions helps us build and understand sequences step by step. This knowledge is super useful in many areas, like calculating money, designing algorithms, and modeling different mathematical scenarios.
Recursive definitions are a cool way to describe sequences. They let us create new terms using the ones that came before. This method helps us get a better grip on different kinds of sequences, especially when you're in Grade 12 Pre-Calculus.
Arithmetic Sequences:
This type uses the rule:
( a_n = a_{n-1} + d )
Here, ( d ) is the common difference.
For example, if we start with ( a_1 = 2 ) and say ( d = 3 ):
The sequence goes like this: ( 2, 5, 8, 11, \ldots )
You can find the total number of terms with this formula:
( a_n = a_1 + (n-1) \cdot d )
Geometric Sequences:
These follow a different rule:
( a_n = a_{n-1} \cdot r )
Here, ( r ) is the common ratio.
For example, if ( a_1 = 3 ) and ( r = 2 ):
The sequence looks like this: ( 3, 6, 12, 24, \ldots )
To find any term in this kind of sequence, you can use:
( a_n = a_1 \cdot r^{n-1} )
Fibonacci Sequences:
In this case, we use the rule:
( F_n = F_{n-1} + F_{n-2} )
with the first two terms as ( F_1 = 1 ) and ( F_2 = 1 ).
This creates the sequence:
( 1, 1, 2, 3, 5, 8, 13, \ldots )
Harmonic Sequences:
These are made from the inverses (or “flips”) of an arithmetic sequence.
For example, starting with ( 1, 2, 3, \ldots ) gives us the harmonic sequence:
( { 1, \frac{1}{2}, \frac{1}{3}, \ldots } )
Using recursive definitions helps us build and understand sequences step by step. This knowledge is super useful in many areas, like calculating money, designing algorithms, and modeling different mathematical scenarios.