Answer: A geometric sequence is a sequence of numbers such that the ratio of any two consecutive terms is constant. Let's call the common ratio "r". We can use this property to find "r" and then calculate other terms in the sequence.
Given that t3 = 24 and t9 = 1536, we can use the formula for the nth term in a geometric sequence: tn = t1 * r^(n-1), where t1 is the first term in the sequence.
Since we know t3 and t9, we can find r by dividing t9 by t3:
r = t9/t3 = 1536/24 = 64
Now that we have found "r", we can use it to find t1 by dividing t3 by r^(3-1):
t1 = t3 / r^(3-1) = 24 / 64^(3-1) = 24 / 64 = 3/2
Now that we know t1 and r, we can find any term in the sequence by using the formula: tn = t1 * r^(n-1).
Therefore, there are two possible sequences with two different first terms:
Sequence 1: t1 = 3/2, r = 64
Sequence 2: t1 = -3/2, r = -64
These are the two possible geometric sequences that satisfy the conditions t3 = 24 and t9 = 1536.
Explanation: