Use a table to organize the terms and find the pattern.
a_1=128, a_n=- 1/4a_(n-1)
Practice makes perfect
To write a recursive rule for the given sequence, we will use a table to organize the terms and find the pattern.
The sequence is geometric, with first term a_1=128 and common ratio r=- 0.25, or - 14.
To write the recursive rule, we will substitute these values into the general formula for the recursive rule of a geometric sequence.
la_1=a a_n=r* a_(n-1) ⇒ la_1= 128 a_n= - 14a_(n-1)
Extra
The Recursive Rule and the Fibonacci Sequence
A recursive rule gives the first term or terms of a sequence and describes how each term is related to the preceding term(s) with a recursive equation. For example, arithmetic and geometric sequences can be described recursively.
Example Arithmetic Sequence
Example Geometric Sequence
Recursive Equation
a_1=3 , a_n = a_(n-1) + 2
a_1=5 , a_n = a_(n-1) * 2
One particularly well-known sequence that is defined recursively is the Fibonacci sequence, in which each term is the sum of the two previous terms. Its first few terms are 1, 1, 2, 3, 5, 8 ... As shown, the first two terms are 1 and 1 and each subsequent term will be the sum of the previous two terms.
For every term after the second term, the sequence can be expressed by the following recursive rule.
&a_1 = 1
&a_2 = 1
&a_n = a_(n - 1) + a_(n - 2).
This means that, in order to determine the fourth term of the sequence, a_4, one must have the terms a_3 and a_2.
a_4 = a_3 + a_2=2+1=3
The sequence is named after the Italian mathematician Leonardo Fibonacci, who used it to describe how pairs of rabbits increases as they multiply under certain conditions. Sometimes, the first two numbers are defined as a_1=0 and a_2=1. This, however, does not change the sequence in any way other than by increasing the index of every term by one compared to the previously mentioned definition.
