Final answer:
To find the explicit equation for the nth term of the geometric sequence, we used the formula an = a1 × rn⁻¹, identifying the first term as 3584 and the common ratio as 0.25. The sixth term was calculated using the derived formula to be 3.5. Equation: an = 3584 × (0.25)n⁻¹
Step-by-step explanation:
The question requires us to find an explicit equation for the nth term of a geometric sequence and then to use it to find the sixth term of that sequence. Our given sequence is 3584, 896, 224, ... which is clearly a geometric sequence where each term is obtained by multiplying the previous term by a common ratio. To find the explicit formula, we need to recognize the common ratio and use the initial term.
The common ratio (r) can be found by dividing any term in the sequence by its preceding term, so 896/3584 = 224/896 = 0.25. The first term (a1) is 3584. With these values, the nth term (an) of the geometric sequence can be found using the formula: an = a1 × rn-1. Plugging in our values, we get an = 3584 × (0.25)n⁻¹.
To find the sixth term, we plug n = 6 into the formula: a6 = 3584 × (0.25)6⁻¹ = 3584 × (0.25)5 = 3584 × 0.0009765625 = 3.5.
Equation: an = 3584 × (0.25)n⁻¹
Sixth term: 3.5