find an explicit rule for the nth term of a geometric sequence where the second and fifth terms are -6 and 162, respectively. a. an = 2 • 3n - 1 b. an = 2 • (-3)n - 1 c. an = 2 • 3n d. an = 2 • (-3)n 1

[tex]a_2=-6\\a_5=162\\\\a_5:a_2=q^3\\\\q^3=162:(-6)\\q^3=-27\\q=\sqrt[3]{-27}\\q=-3\\\\a_n=a_1\cdot q^{n-1}\\\\a_1=a_2:q\to a_1=-6:(-3)=a_1=2\\\\\boxed{a_n=2\cdot(-3)^{n-1}\to b.}[/tex]

Answer Link

johanrusli johanrusli · Answer 2 · 2019-07-09T10:42:55+02:00

An explicit rule for the n - th term of a geometric sequence where the second and fifth terms are -6 and 162, respectively is b. Tn = 2 • (-3)ⁿ⁻¹

Further explanation

Firstly , let us learn about types of sequence in mathematics.

Arithmetic Progression is a sequence of numbers in which each of adjacent numbers have a constant difference.

[tex]\boxed {T_n = a + (n-1)d}[/tex]

[tex]\boxed {S_n = \frac{1}{2}n ( 2a + (n-1)d )}[/tex]

Tn = n-th term of the sequence

Sn = sum of the first n numbers of the sequence

a = the initial term of the sequence

d = common difference between adjacent numbers

Geometric Progression is a sequence of numbers in which each of adjacent numbers have a constant ratio.

[tex]\boxed {T_n = a ~ r^{n-1}}[/tex]

[tex]\boxed {S_n = \frac{a( 1 - r^n ) }{1 - r}}[/tex]

Tn = n-th term of the sequence

Sn = sum of the first n numbers of the sequence

a = the initial term of the sequence

r = common ratio between adjacent numbers

Let us now tackle the problem!

Given:

T₂ = -6

T₅ = 162

Unknown:

Tn = ?

Solution:

[tex]T_2 = a ~ r^{2-1}[/tex]

[tex]-6 = a ~ r[/tex]

[tex]a = \frac{-6}{r}[/tex] → Equation ( 1 )

[tex]T_5 = a ~ r^{5-1}[/tex]

[tex]162 = a ~ r^4[/tex]

[tex]162 = \frac{-6}{r} \times r^4[/tex] ← Equation ( 1 )

[tex]162 = \frac{-6}{r} \times r^4[/tex]

[tex]\frac{162}{-6} = r^3[/tex]

[tex]r^3 = -27[/tex]

[tex]r = \sqrt [3]{-27}[/tex]

[tex]r = -3[/tex]

[tex]a = \frac{-6}{r}[/tex] ← Equation ( 1 )

[tex]a = \frac{-6}{-3}[/tex]

[tex]a = 2[/tex]

Finally , the explicit rule for the n - th term is :

[tex]T_n = a ~ r^{n-1}[/tex]

[tex]\large { \boxed {T_n = 2 \cdot (-3)^{n-1} } }[/tex]

Learn more

Geometric Series : https://brainly.com/question/4520950
Arithmetic Progression : https://brainly.com/question/2966265
Geometric Sequence : https://brainly.com/question/2166405

Answer details

Grade: High School

Subject: Mathematics

Chapter: Arithmetic and Geometric Series

Keywords: Arithmetic , Geometric , Series , Sequence , Difference , Term