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Find the common ratio & the next three terms in the sequence.
2.-6, 18, -54, ...
a
Common Ratio: r= 3 Next 3 terms: 32, -64, 128
b
Common Ratio: r = 5 Next 3 terms: 1250, 6250, 31250
C
Common Ratio: r = 5 Next 3 terms: 250, 1250, 6250
Common Ratio: r = -B Next B terms: 162, -486, 1458

Sagot :

Absor201idn

Answer:

We conclude that:

  • Common ratio: r = -3
  • Next three terms: 162, -486, 1458

Hence, option D is correct.

Step-by-step explanation:

Given the geometric sequence

2, -6, 18, -54, ...

Here:

[tex]a_1=2[/tex]

A geometric sequence has a constant difference 'r' and is defined by  

[tex]a_n=a_1\cdot \:r^{n-1}[/tex]

computing the differences of all the adjacent terms

[tex]\frac{-6}{2}=-3,\:\quad \frac{18}{-6}=-3,\:\quad \frac{-54}{18}=-3[/tex]

The ratio of all the adjacent terms is the same and equal to

[tex]r=-3[/tex]

now substituting [tex]a_1=2[/tex] and [tex]r=-3[/tex] in the nth term of the sequence

[tex]a_n=a_1\cdot \:r^{n-1}[/tex]

now substituting n = 5 to determine the 5th term

[tex]a_5=2\left(-3\right)^{5-1}[/tex]

[tex]a_5=3^4\cdot \:2[/tex]

[tex]a_5=2\cdot \:81[/tex]

[tex]a_5=162[/tex]

now substituting n = 6 to determine the 6th term

[tex]a_6=2\left(-3\right)^{6-1}[/tex]

[tex]a_6=-2\cdot \:243[/tex]

[tex]a_6=-486[/tex]

now substituting n = 7 to determine the 7th term

[tex]a_7=2\left(-3\right)^{7-1}[/tex]

[tex]a_7=2\cdot \:729[/tex]

[tex]a_7=1458[/tex]

Therefore, we conclude that:

  • Common ratio: r = -3
  • Next three terms: 162, -486, 1458

Hence, option D is correct.

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