The first three terms of a geometric sequence are 2, 10, 50, and the 8th term has a value of 156,250. What is the value of the 7th term?

To find the value of the 7th term in a geometric sequence, we first need to determine the common ratio between each consecutive term.

In a geometric sequence, each term is found by multiplying the previous term by a fixed number called the common ratio (r).

We can find the common ratio (r) by dividing the second term (10) by the first term (2):

r = (2nd term) / (1st term) = 10 / 2 = 5.

Now that we have found the common ratio (r), we can find the 7th term using the formula for a geometric sequence:

T(n) = a * r^(n-1),

where T(n) is the nth term of the sequence, a is the first term, r is the common ratio, and n is the position of the term.

In this case, we want to find the 7th term, so n = 7. The formula becomes:

T(7) = a * r^(7-1) = a * r^6.

Using the value of the first term (a = 2) and the common ratio (r = 5), we can calculate the 7th term:

T(7) = 2 * 5^6 = 2 * 15625 = 31250.

Therefore, the value of the 7th term in the sequence is 31,250.