Answer:
see explanation
Step-by-step explanation:
The n th term of a geometric sequence is
[tex]a_{n}[/tex] = a[tex]r^{n-1}[/tex]
where a is the first term and r the common ratio, thus
[tex]a_{3}[/tex] = ar² = 12 → (1)
[tex]a_{5}[/tex] = a[tex]r^{4}[/tex] = 48 → (2)
Divide (2) by (1)
[tex]\frac{ar^4}{ar^2}[/tex] = [tex]\frac{48}{12}[/tex] = 4
r² = 4 ⇒ r = ± 2 ⇒ r = 2 ← since r > 0
Substitute r = 2 into (1) for corresponding value of a
4a = 12 ⇒ a = 3
Hence
[tex]a_{n}[/tex] = 3[tex](2)^{n-1}[/tex] ← explicit rule
