Question

Which formula can be used to find the nth term of the following geometric sequence? − 2/9 , 2/3 , −2, 6,…

Answer 1

Answer:

aₙ = (-1)ⁿ2(3)ⁿ⁻³

Step-by-step explanation:

The formula for the nth term of a geometric sequence is

aₙ = a₁rⁿ⁻¹

We can get the value for r by dividing aₙ by aₙ₋₁.

a₄/a₃ = 6 ÷ (-2)

a₄/a₃ = -3

r = -3

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a₁ = -2/9

aₙ = (-2/9)(-3)ⁿ⁻¹

aₙ = (-2/3²)(-3)ⁿ⁻¹              Factor out the 3s from the 9

aₙ = -2(3²)(-1)ⁿ⁻¹(3)ⁿ⁻¹        Factor -1 from  and -3

aₙ  = (-1)(2)(-1)ⁿ⁻¹(3)ⁿ⁻³      Combine like terms

aₙ = 2(-1)ⁿ⁻²(3)ⁿ⁻³             Multiply the -1 term by (-1)² (=1)

aₙ = (-1)ⁿ2(3)ⁿ⁻³

The (-1)ⁿ term makes the terms alternately positive and negative.

Answer 2

Answer:

$a_n=-\frac{2}{9}(-3)^{n-1}$

Step-by-step explanation:

Since, the nth term of a geometric sequence is,

$a_n=ar^{n-1}$

Where, a is the first term of the sequence,

r is the common ratio,

Here, the given geometric sequence is,

$-\frac{2}{9},\frac{2}{3},-2,6........$

Thus, the first term of the sequence is,

$a=-\frac{2}{9}$

And, the common ratio,

$r=\frac{2/3}{-2/9}=-3$

Hence, the nth term of the sequence is,

$a_n=-\frac{2}{9}(-3)^{n-1}$