The sum of the first 3 terms of a geometric series is 26. The sum of the first 6 terms is 728. Find the first term and common ratio.

Let's denote the first term of the geometric series as 'a' and the common ratio as 'r'.

From the problem, we know that:

1. a + ar + ar^2 = 26 (This is the sum of the first 3 terms)

2. a + ar + ar^2 + ar^3 + ar^4 + ar^5 = 728 (This is the sum of the first 6 terms)

We can simplify the second equation by subtracting the first equation from it:

ar^3 + ar^4 + ar^5 = 728 - 26 = 702

We can factor out ar^3 from the left side:

ar^3(1 + r + r^2) = 702

But we know from the first equation that 1 + r + r^2 = 26, so we can substitute:

ar^3 * 26 = 702

Solving for ar^3 gives us:

ar^3 = 702 / 26 = 27

Now we have two equations:

1. a + ar + ar^2 = 26

2. ar^3 = 27

We can solve these equations simultaneously to find the values of 'a' and 'r'.

From the second equation, we can express 'a' as a = 27 / r^3.

Substitute 'a' into the first equation:

27/r^3 + 27/r^2 + 27/r = 26

Multiply through by r^3 to clear the denominators:

27 + 27r + 27r^2 = 26r^3

Rearrange to form a cubic equation:

26r^3 - 27r^2 - 27r - 27 = 0

This is a cubic equation in 'r' which can be solved using methods for solving cubic equations. Once you find 'r', you can substitute it back into the equation a = 27 / r^3 to find 'a'.

Note: The cubic equation may have more than one root, but in the context of a geometric series, 'r' should be a real number other than 0 or 1.