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What is the explicit rule for this geometric sequence?

a₁=−15; aₙ=1/5 ⋅ aₙ₋₁
a. aₙ = -15(1/5)ⁿ⁻¹
b. aₙ = 1/5 . (-15)ⁿ⁻¹
c. aₙ = 15(-1/5)ⁿ⁻¹
d. aₙ = -1/5 . 15ⁿ⁻¹

1 Answer

6 votes

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.

User Arychj
by
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