Final answer:
To determine the common ratio of the geometric sequence with a1=2400 and a4=37.5, we calculate the third root of the quotient 37.5/2400, resulting in the common ratio of 0.25.
Step-by-step explanation:
To find the common ratio of a geometric sequence given two terms, you can use the formula for the nth term of a geometric sequence, which is a_n = a_1 × r^{(n-1)}, where a_n is the nth term, a_1 is the first term, and r is the common ratio. The student provided the first term as a_1 = 2400 and the fourth term as a_4 = 37.5.
We can use these two terms to solve for the common ratio as follows:
- Set up the equation using the formula for the nth term: 37.5 = 2400 × r^{(4-1)}
- Solve for r by first dividing both sides by 2400: r^{3} = 37.5/2400
- Calculate the result on the right side: r^{3} = 0.015625
- Find the cube root of both sides to get the common ratio: r = ∛0.015625
- The common ratio is r = 0.25.
Therefore, the common ratio for the geometric sequence is 0.25.