Question

Write the explicit formula for the geometric sequence represented by the recursive formula.

a1 = 4
an = 5 · an-1
A) an = 4 · 5n
B) an = 5 · 4n
C) an = 4 · 5n-1
D) an = 5 · 4n-1

Answer 1

The answer is c - an = 4*5n-1

Answer 2

Option C is correct.

`a_n =4 \cdot 5^(n-1)`

Step-by-step explanation:

Explicit formula for the geometric sequence is given by:

`a_n =a_1 \cdot r^(n-1)`

where r is the common ratio

Given the recursive formula for geometric sequence:

`a_1 = 4`

`a_n =5 \cdot a_(n-1)`

For n =2

`a_2 = 5 \cdot a_(2-1)`

`a_2 = 5 \cdot a_1`

⇒
`a_2 = 5 \cdot 4 = 20`

For n =3

`a_3= 5 \cdot a_(3-1)`

`a_3= 5 \cdot a_(2)`

⇒
`a_3= 5 \cdot 20 = 100`

Common ratio(r):

`(a_2)/(a_1) = (20)/(4) = 5`

`(a_3)/(a_2) = (100)/(20) = 5`

⇒ r = 5

Therefore, the explicit formula for the geometric sequence represented by the recursive formula is:

`a_n =4 \cdot 5^(n-1)`