The recursive rule for a geometric sequence is given. a1=2;an=13an−1

The complete question is:
The recursive rule for a geometric sequence is given. [tex] a_{1}=2 [/tex]; [tex] a_{n}= 13*a_{n-1} [/tex]. Enter the explicit rule for the sequence.

Solution:
We are given
[tex] a_{1}=2 \\ \\ a_{n} = a_{n-1} [/tex]

Explicit rule of geometric sequence is of the form:

[tex] a_{n} = a_{1} (r)^{n-1} [/tex]

Using the given recursive sequence, we can find r.

[tex] a_{1}=2 [/tex]
[tex] a_{2}=13* a_{1}=13*2=26 [/tex]

r = Ratio of consecutive two terms of Geometric series.
So,
r = 26/2 = 13

Therefore,

[tex] a_{n}=2(13)^{n-1} [/tex] is the explicit rule for the given geometric sequence.