Respuesta :
Answer:
Option C is correct.
[tex]a_n =4 \cdot 5^{n-1}[/tex]
Explanation:
Explicit formula for the geometric sequence is given by:
[tex]a_n =a_1 \cdot r^{n-1}[/tex]
where r is the common ratio
Given the recursive formula for geometric sequence:
[tex]a_1 = 4[/tex]
[tex]a_n =5 \cdot a_{n-1}[/tex]
For n =2
[tex]a_2 = 5 \cdot a_{2-1}[/tex]
[tex]a_2 = 5 \cdot a_1[/tex]
⇒[tex]a_2 = 5 \cdot 4 = 20[/tex]
For n =3
[tex]a_3= 5 \cdot a_{3-1}[/tex]
[tex]a_3= 5 \cdot a_{2}[/tex]
⇒[tex]a_3= 5 \cdot 20 = 100[/tex]
Common ratio(r):
[tex]\frac{a_2}{a_1} = \frac{20}{4} = 5[/tex]
[tex]\frac{a_3}{a_2} = \frac{100}{20} = 5[/tex]
⇒ r = 5
Therefore, the explicit formula for the geometric sequence represented by the recursive formula is:
[tex]a_n =4 \cdot 5^{n-1}[/tex]
