Final answer:
To find the sum of the first five terms of a geometric series with given second and fifth terms, calculate the common ratio and the first term of the series, then apply the sum formula for a geometric series.
Step-by-step explanation:
The question asks to find the sum of the first five terms of a geometric series. Given a₂ (the second term) is -12 and a₅ (the fifth term) is 768, we can use the formula for the nth term of a geometric series, which is aᵢ = a₁r¹⁸⁴, where aᵢ is the nth term, a₁ is the first term, and r is the common ratio. To find the common ratio, we use the fact that a₅ = a₁r⁴. Dividing a₅ by a₂ gives us r³, and we can then calculate r.
Once r is found, we calculate a₁ from a₂ = a₁r. With a₁ and r, we can find the sum of the first five terms using the formula for the sum of a geometric series: Sₙ = a₁(1 - rₙ)/(1 - r) for n terms.
This process will lead us to the correct answer, which should match one of the provided options in the question.