Given the sequence 3, 6, 12, 24, 48, 96, ....
F. Write an explicit formula for the geometric sequence. Use your explicit formula to find the 32nd term of the sequence.
To find the explicit formula for the geometric sequence, we need to determine the common ratio. We can do this by dividing any term by its preceding term. Let's take the second term, 6, and divide it by the first term, 3:
6/3 = 2
The common ratio is 2. We can now use this common ratio to write the explicit formula for the geometric sequence:
a_n = a_1 * r^(n-1)
where a_n is the nth term of the sequence, a_1 is the first term, r is the common ratio, and n is the position of the term.
Given that the first term, a_1, is 3 and the common ratio, r, is 2, we can now find the 32nd term, a_32:
a_32 = 3 * 2^(32-1)
a_32 = 3 * 2^31
a_32 = 3 * 2,147,483,648
a_32 = 6,442,451,944