Respuesta :
To find the explicit formula of geometric sequences, you'll need to find a formula for the nth term.
In symbols, the nth term of a geometric sequence is: tn = a·rn-1.
a = first term and r = common ratio
To find the common ratio, divide any term by its preceding term.
Example: 2, 6, 18, 54, 162, ...
a = first term = 2
r = common ratio = 6/3 = 2 (this will be the same anywhere you begin: 162/54 = 3, 54/18 = 3, 18/6 = 3, etc.)
So, the explicit formula is: tn = 2·3n-1
Each explicit formula will have the exponent "n-1".
Your answer would be; tn = 2·3n-1
The explicit rule for this geometric sequence is [tex]\boxed{2 \times {{\left( 3 \right)}^{n - 1}}}[/tex]. Option (c) is correct.
Further Explanation:
If the first term a and the second term ar is known then, the value of r can be obtained as follows,
[tex]\boxed{r =\dfrac{{{a_2}}}{{{a_1}}}}[/tex]
The nth term of the geometric sequence can be obtained as,
[tex]\boxed{{a_n} = a \times {r^{n - 1}}}[/tex] …… (1)
Given:
The sequence is [tex]2, 6, 18, 54[/tex],…
The options are as follows,
(a).[tex]{a_n} = 3 \cdot 2n - 1[/tex]
(b).[tex]{a_n} = 2 \cdot 3n[/tex]
(c).[tex]{a_n} = 2 \cdot 3n - 1[/tex]
(d).[tex]{a_n} = 3 \cdot 2n[/tex]
Explanation:
The first term of the sequence is 2, second term of the sequence, is 6, third term is 18.
The common ratio r can be obtained as follows.
[tex]\begin{aligned}r&=\frac{{{a_2}}}{{{a_1}}}\\&=\frac{6}{2}\\&= 3\\\end{aligned}[/tex]
The common ratio is 3.
Substitute 3 for r and 2 for a in equation (1) to obtain the nth term.
[tex]{a_n} = 2 \times {\left( 3 \right)^{n - 1}}[/tex]
The explicit rule for this geometric sequence is [tex]\boxed{2 \times {{\left( 3 \right)}^{n - 1}}}.[/tex] Option (c) is correct.
Learn more:
1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.
2. Learn more about equation of circle brainly.com/question/1506955.
3. Learn more about range and domain of the function https://brainly.com/question/3412497
Answer details:
Grade: High School
Subject: Mathematics
Chapter: Geometric progression
Keywords: geometric sequence, explicit rule, common ratio, first term, second term, sum of geometric sequence, nth term of the geometric sequence.
