We will find the first four terms of the described sequence, then graph them. After that, we will write a recursive and an explicit rule for the sequence.
Graph
We will make a table to show the first four terms. Note that the first term is - 1 and the common ratio is - 3.
Let's plot the ordered pairs (1,- 1), (2,3), (3,- 9), and (4,27).
Recursive Rule
The sequence a_n is a geometric sequence, with first term a_1=- 1 and common ratio r= - 3. We can write the recursive equation.
a_n= r* a_(n-1) ⇒ a_n= - 3* a_(n-1)
The recursive rule is the recursive equation together with the first term.
Recursive Rule: a_1=- 1, a_n=- 3* a_(n-1)
Explicit Rule
The explicit rule for a geometric sequence is the formula a_n=a_1(r)^(n-1), where a_1 is the first term and r is the common ratio. We know a_1= - 1 and r= - 3
a_n= a_1( r)^(n-1) ⇒ a_n= - 1( - 3)^(n-1)
Since multiplying any number by 1 does not change the result, we can remove it.
Explicit Rule: a_n=- (- 3)^(n-1)
