Answer:
a[n] = a[n-1]×(4/3)
a[1] = 1/2
Explanation:
The terms of a geometric sequence have an initial term and a common ratio. The common ratio multiplies the previous term to get the next one. That sentence describes the recursive relation.
The general explicit term of a geometric sequence is ...
a[n] = a[1]×r^(n-1) . . . . . where a[1] is the first term and r is the common ratio
Comparing this to the expression you are given, you see that ...
a[1] = 1/2
r = 4/3
(You also see that parenthses are missing around the exponent expression, n-1.)
A recursive rule is defined by two things:
- the starting value(s) for the recursive relation
- the recursive relation relating the next term to previous terms
The definition of a geometric sequence tells you the recursive relation is:
the next term is the previous one multiplied by the common ratio.
In math terms, this looks like
a[n] = a[n-1]×r
Using the value of r from above, this becomes ...
a[n] = a[n-1]×(4/3)
Of course, the starting values are the same for the explicit rule and the recursive rule:
a[1] = 1/2