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Consider the following geometric sequence:

[tex] \[ 115, 690, 4, 140, 24, 840, \ldots \] [/tex]

What is the explicit formula for this sequence?

A. [tex] \[ a_n = 115 \cdot \left(\frac{1}{6}\right)^{(n-1)} \] [/tex]

B. [tex] \[ a_n = 115 \cdot (-6)^{(n-1)} \] [/tex]

C. [tex] \[ a_n = 115 \cdot 6^{(n-1)} \] [/tex]

D. [tex] \[ a_n = 115 \cdot 5^{(n-1)} \] [/tex]

Answer :

To find the explicit formula for the given geometric sequence, let's break down the steps:

1. Identify the First Term:
The first term of the sequence is [tex]\( a_1 = 115 \)[/tex].

2. Determine the Common Ratio:
In a geometric sequence, each term is obtained by multiplying the previous term by a constant called the common ratio ([tex]\( r \)[/tex]).
To find [tex]\( r \)[/tex], you calculate the ratio between the second term and the first term:
[tex]\[
r = \frac{690}{115} = 6
\][/tex]
This tells us that each term is 6 times the previous term.

3. Write the Explicit Formula:
The explicit formula for a geometric sequence is given by:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
Substituting the known values ([tex]\( a_1 = 115 \)[/tex] and [tex]\( r = 6 \)[/tex]) into the formula, we get:
[tex]\[
a_n = 115 \times 6^{(n-1)}
\][/tex]

Therefore, the explicit formula for this geometric sequence is [tex]\( a_n = 115 \cdot 6^{(n-1)} \)[/tex]. This matches one of the given options:
- [tex]\( a_n=115 \cdot(6)^{(n-1)} \)[/tex]

This is the correct choice.

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