Question

What is the explicit formula for aₙ, the nth term of the sequence 32, -16, 8...? a) aₙ = 32 x (-1)ⁿ⁻¹ b) aₙ = 32 x (-2)ⁿ⁻¹ c) aₙ = 32 x (-2)ⁿ d) aₙ = 32 x (-1)ⁿ

Answer 1

The explicit formula for the nth term of the given sequence is a₍ = 32 × (-1)ⁿ¹⁽ⁱ, derived from the properties of a geometric sequence with the first term of 32 and a common ratio of -1/2.

Step-by-step explanation:

The explicit formula for the nth term of the sequence 32, -16, 8... is a₍ = 32 × (-1)ⁿ¹⁽ⁱ. This sequence is geometric, with each term being multiplied by -1/2 to get the next term. The first term (a₁) is 32, and the common ratio (r) is -1/2. The formula for any term in a geometric sequence is a₍ = a₁ × rⁿ¹. Substituting the specific values gives us the explicit formula for the nth term: a₍ = 32 × (-1/2)ⁿ¹⁽ⁱ = 32 × (-1)ⁿ¹⁽ⁱ.