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:
[tex]a_n=-\frac{2}{9}(-3)^{n-1}[/tex]
Step-by-step explanation:
Since, the nth term of a geometric sequence is,
[tex]a_n=ar^{n-1}[/tex]
Where, a is the first term of the sequence,
r is the common ratio,
Here, the given geometric sequence is,
[tex]-\frac{2}{9},\frac{2}{3},-2,6........[/tex]
Thus, the first term of the sequence is,
[tex]a=-\frac{2}{9}[/tex]
And, the common ratio,
[tex]r=\frac{2/3}{-2/9}=-3[/tex]
Hence, the nth term of the sequence is,
[tex]a_n=-\frac{2}{9}(-3)^{n-1}[/tex]
