Answer :
Final answer:
To find the four geometric means between 1 and 3125, you first determine the common ratio of the geometric sequence by calculating 3125^(1/5). The ratio is 5. Then, you calculate the four geometric means by multiplying 1 by 5, 5^2, 5^3, and 5^4, resulting in 5, 25, 125, and 625.
Explanation:
In a geometric sequence, we can find the geometric means between two numbers by using the formula for the nth term of a geometric sequence, which is a * r^(n-1), where 'a' is the first term, 'r' is the common ratio, and 'n' is the term number.
In this case, the first term a is 1, and the 6th term (n=6) is 3125. We have 1 * r^5 = 3125, that gives us r = 3125^(1/5) = 5.
Now, we can calculate the geometric means by substituting n = 2, 3, 4, 5 into the formula to get the 2nd, 3rd, 4th, 5th term of the sequence. So, the four geometric means between 1 and 3125 are 1*5, 1*5^2, 1*5^3, 1*5^4, which are 5, 25, 125, and 625.
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