Final answer:
The explicit rule for this geometric sequence is aₙ = -15(1/5)ⁿ⁻¹. option A is correct answer.
Step-by-step explanation:
The explicit rule for this geometric sequence is:
an = -15(1/5)n-1
Step-by-step explanation:
The first term of the sequence, a1, is given as -15.
To find the nth term, an, we can use the formula an = a1 * rn-1, where r is the common ratio.
In this case, the common ratio is 1/5.
Substituting the values into the formula, we get an = -15 * (1/5)n-1.
The given geometric sequence has a first term
�
1
=
−
15
a
1
=−15 and a recursive formula
�
�
=
1
5
⋅
�
�
−
1
a
n
=
5
1
⋅a
n−1
where
�
n represents the term number. To find the explicit rule, we need to determine the common ratio (
�
r) of the sequence.
Since
�
�
=
1
5
⋅
�
�
−
1
a
n
=
5
1
⋅a
n−1
, the common ratio
�
r is
1
5
5
1
. Now, we can write the explicit rule for a geometric sequence:
�
�
=
�
1
⋅
�
(
�
−
1
)
a
n
=a
1
⋅r
(n−1)
Substitute the values:
�
�
=
−
15
⋅
(
1
5
)
(
�
−
1
)
a
n
=−15⋅(
5
1
)
(n−1)
This is the explicit rule for the given geometric sequence. It allows you to directly find any term
�
�
a
n
without having to recursively calculate the preceding terms. The negative sign in front of 15 indicates that each term alternates in sign between positive and negative.