Answer:
The Recursive Formula for the sequence is:
[tex]\:a_n=\frac{1}{5}\left(a_{n-1}\right)[/tex] ; a₁ = 125
Hence, option D is correct.
Step-by-step explanation:
We know that a geometric sequence has a constant ratio 'r'.
The formula for the nth term of the geometric sequence is
[tex]a_n=a_1\cdot r^{n-1}[/tex]
where
aₙ is the nth term of the sequence
a₁ is the first term of the sequence
r is the common ratio
We are given the explicit formula for the geometric sequence such as:
[tex]a_n=125\left(\frac{1}{5}\right)^{n-1}[/tex]
comparing with the nth term of the sequence, we get
a₁ = 125
r = 1/5
Recursive Formula:
We already know that
We know that each successive term in the geometric sequence is 'r' times the previous term where 'r' is the common ratio.
i.e.
[tex]a_n=ra_{n-1}[/tex]
Thus, substituting r = 1/5
[tex]\:a_n=\frac{1}{5}\left(a_{n-1}\right)[/tex]
and a₁ = 125.
Therefore, the Recursive Formula for the sequence is:
[tex]\:a_n=\frac{1}{5}\left(a_{n-1}\right)[/tex] ; a₁ = 125
Hence, option D is correct.
