Write the geometric sequence that has four geometric means between 1 and 16,807.

r^5 = 16807

r = 7

Now just write the terms

Find the sum of the terms of a geometric seqence where the first term is 1,the last term is 16 807 and the common ratko is 7.

To find the geometric sequence with four geometric means between 1 and 16,807, we need to find the common ratio (r) and the first term (a).

Let's form the equation using the given information:

16,807 = a * r^6 (since there are four geometric means, we raise the common ratio to the power of 6)
1 = a * r^2 (there are six terms between the first term and the last term, so we raise the common ratio to the power of 2)

To eliminate 'a', we divide the two equations:

(16,807 / 1) = (a * r^6) / (a * r^2)
16,807 = r^4

Now we can solve for 'r':

r = ∛(16,807)
r = 13

Now we can find 'a' using the second equation:

1 = a * (13^2)
1 = 169a
a = 1/169

Therefore, the geometric sequence with four geometric means between 1 and 16,807 is:

1/169, 1/13, 1, 13, 169, 2197, 28561, 371293, 4826809

To find the geometric sequence with four geometric means between 1 and 16,807, we need to determine the common ratio (r) of the sequence and then write out the terms.

Let's start by finding the common ratio (r). In a geometric sequence, each term is obtained by multiplying the previous term by the common ratio.

Here, the first term (a) is 1 and the last term (an) is 16,807. We need to determine the ratio r.

Using the formula for the nth term of a geometric sequence:
an = a * r^(n-1)

Set up the equation using the provided values:
16,807 = 1 * r^(6-1)

Now, let's solve for r:
16,807 = r^5

To isolate r, we can take the fifth root of both sides of the equation:
r = 16,807^(1/5)

Calculating this value, we find that r is approximately 7.

Now, we have the common ratio (r = 7) and the first term (a = 1).

To write out the geometric sequence, we have:
1, 7, 49, 343, 2401, 16,807

These are the terms of the geometric sequence that has four geometric means between 1 and 16,807.