It is a formula, used in sequences, in which the value of the nth term is described in relation to one or more of the earlier terms.
A classic example is the Fibonacci sequence:
u(1) = 1
u(2) = 1
u(n) = u(n-1) + u(n-2) for n = 3, 4, 5, ...
what is the recursive formula for this geometric sequence?
It is often possible to find an explicit formula that gives the same answer as a given recursive formula - and vice versa. I don't think you can always find an explicit formula that gives the same answer.
The sequence 1, 4, 13, 40, 121 can be described by a recursive formula. The recursive relationship can be expressed as ( a_n = 3a_{n-1} + 1 ) for ( n \geq 2 ), with the initial condition ( a_1 = 1 ). This means each term is generated by multiplying the previous term by 3 and then adding 1.
In general, it is not possible to uniquely determine a recursive rule or formula with only the first two terms of a sequence. While the initial terms can suggest a pattern, multiple recursive sequences can produce the same first two terms. To accurately derive a recursive rule, additional terms are typically needed to identify the underlying pattern or relationship governing the sequence.
That sounds like the definition for Recursive Formula.
what is the recursive formula for this geometric sequence?
It is often possible to find an explicit formula that gives the same answer as a given recursive formula - and vice versa. I don't think you can always find an explicit formula that gives the same answer.
The recursive formula for the function f(n) is f(n) f(n/2).
To input a recursive equation into a TI-84 calculator, you can use the "Seq" function. First, access the "Y=" menu, then define your recursive sequence by using the format Seq(Y1, n, start, end), where Y1 is your recursive formula, and "start" and "end" are the range of values for n. Alternatively, you can manually calculate the terms by iterating through the recursive formula using the calculator's programming feature or list functions.
It look like a Fibonacci sequence seeded by t1 = 2 and t2 = 1. After that the recursive formula is simply tn+1 = tn-1 + tn.
x_n+1 = x_n / 4
A recursive formula is one that references itself. The famous example is the Fibonacci function: fib(n) := fib(n-1) + fib(n-2), with the terminating proviso that fib(0) = 0 and fib(1) = 1.
The sequence 1, 4, 13, 40, 121 can be described by a recursive formula. The recursive relationship can be expressed as ( a_n = 3a_{n-1} + 1 ) for ( n \geq 2 ), with the initial condition ( a_1 = 1 ). This means each term is generated by multiplying the previous term by 3 and then adding 1.
In general, it is not possible to uniquely determine a recursive rule or formula with only the first two terms of a sequence. While the initial terms can suggest a pattern, multiple recursive sequences can produce the same first two terms. To accurately derive a recursive rule, additional terms are typically needed to identify the underlying pattern or relationship governing the sequence.
Recursive and explicit rules are both methods used to define sequences in mathematics. They both provide a way to generate terms of a sequence, where a recursive rule defines each term based on previous terms, while an explicit rule provides a formula to calculate any term directly. Despite their different approaches, both types of rules ultimately serve the same purpose: to describe the pattern or relationship within a sequence. Additionally, both can be used to analyze and predict future terms in the sequence.
That sounds like the definition for Recursive Formula.
4, -1236, -108 is not a geometric system.