Identify the 12th term of a geometric sequence where a₁ = 8 and a₆ = -8,192.

Solution:

The n^th term of a geometric sequence is given by

a_n = a₁ × r^{n - 1} ---------> (1)

Given, a₁ = 8, a₆ = -8192, n = 12

To find r put the value of a₁ and a₆ in (1)

-8192 = 8 × r^{(6 - 1)}

r⁵ = -8192/8

r⁵ = 1024

r = \(\sqrt[5]{1024}\)

r = -4

Now, find a₁₂

a₁₂ = a₁ × r¹¹

a₁₂ = 8 × (-4)⁷

= 8 × (-16,384)

a₁₂ = -131072

Therefore, the 12^th term of the series is -131072.

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Identify the 12th term of a geometric sequence where a₁ = 8 and a₆ = -8,192.

Summary:

The12th term of the geometric sequence where a₁ = 8 and a₆ = -8192 is -131072.