Question

Will give brainliest-What is the explicit rule for this geometric sequence?

2, 6, 18, 54, …



an=3⋅2n−1

an=2⋅3n

an=2⋅3n−1

an=3⋅2n

Answer 1

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

Answer 2

The explicit rule for this geometric sequence is $\boxed{2\times{{\left(3\right)}^{n-1}}}$.

Further Explanation:

The terms of the geometric sequence can be written as,

$a,ar,a{r^2},a{r^3},..$

Here, a is the first term and r is the common ratio.

If the first term a and the second term ar is known then, the value of r can be obtained as follows,

$\boxed{r=\frac{{{a_2}}}{{{a_1}}}}$

The $nth$ term of the geometric sequence can be obtained as,

$\boxed{{a_n}=a\times{r^{n-1}}}$                                                                …… (1)

Now, the value of any term can be easily obtained with the help of $a$ and $r$.

Given:

The sequence is $2,6,18,54,....$

Explanation:

The given sequence is $2,6,18,54,.....$

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.

$\begin{aligned}r&=\frac{{{a_2}}}{{{a_1}}}\\&=\frac{6}{2}\\&=3\\\end{aligned}$

The common ratio is 3.

Substitute 3 for $r$ and 2 for $a$ in equation (1) to obtain the $nth$ term.

${a_n}=2\times{\left(3\right)^{n-1}}$

The explicit rule for this geometric sequence is $\boxed{2\times{{\left(3\right)}^{n-1}}}$.

Learn more:

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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.