Final answer:
To find the common ratio and the first term of a geometric sequence, we used two known terms and the formula for the nth term of a geometric sequence. By dividing the seventh term by the fourth term, we deduced that the common ratio is 2. Then, we used the common ratio to find the first term, which is 6.
Step-by-step explanation:
In the given problem, we need to find the common ratio (r) and the first term (a₁) of a geometric sequence given that the fourth term (a₄) is 48, and the seventh term (a₇) is 384. Recall that any term in a geometric sequence can be found using the formula aₙ = a₁ × r^(n-1), where aₙ is the nth term, a₁ is the first term, r is the common ratio, and n is the term number.
Using the information for the fourth and seventh terms, we can write two equations:
- a₄ = a₁ × r^(4-1) = a₁ × r^3 = 48
- a₇ = a₁ × r^(7-1) = a₁ × r^6 = 384
Then, we divide the second equation by the first to eliminate a₁ and solve for r:
(384 / 48) = (a₁ × r^6) / (a₁ × r^3)\
8 = r^3\\
r = ∛8 = 2
Substituting r = 2 into the equation for a₄:
48 = a₁ × 2^3\\
48 = a₁ × 8\\
a₁ = 48 / 8\\
a₁ = 6
Therefore, the common ratio (r) is 2, and the first term (a₁) is 6. The correct answer is d) r=2, a₁=6.