We are given two non-consecutive of a .
a2=-72anda6=-181
To find the rule for the
nth term, we first need to find the
r. Let's start by recalling the of a geometric sequence.
an=a1rn−1
We will substitute the given information into this formula to write a . By solving the system, we will find the values of
r and
a1.
{a2=a1r2−1a6=a1r6−1⇒{-72=a1r-181=a1r5(I)(II)
To solve the system, we will start by dividing Equation (II) by Equation (I). By doing this, we will eliminate
a1 and find the value of
r.
-72-181=a1ra1r5
72181=a1ra1r5
72(18)1=a1ra1r5
12961=a1ra1r5
12961=rr5
12961=r5−1
12961=r4
61=±r
r=±61
We found out that
r has two possible solutions,
-61 and
61. Knowing the values of
r, we can substitute them into either of the equations of our system in order to find the value of
a1.
First Solution
Let's substitute
r=-61 in Equation (I).
-72=a1r
-72=a1(-61)
-72=-a1(61)
-432=-a1
-432+a1=0
a1=432
Finally, we can write the complete formula for the first solution.
an=a1rn−1⇒an=432(-61)n−1
Second Solution
Let's substitute
r=61 in Equation (I).
-72=a1r
-72=a1(61)
-432=a1
a1=-432
Finally, we can write the complete formula for the second solution.
an=a1rn−1⇒an=-432(61)n−1