In an arithmetic series consisting of 51 terms, the sum of the first three terms is 65 and the sum of the middle three terms is 129. Find out the first term and the common difference of the series.

Sure, let's solve this step by step.

1. In an arithmetic series, the middle term is the average of the first and the last term. So, we can write the equation as:

   (a + a + 50d) / 2 = 129 / 3 => 2a + 50d = 43 * 3 => 2a + 50d = 129 ---- (1)

2. The sum of the first three terms is 65, which can be written as:

   a + (a + d) + (a + 2d) = 65 => 3a + 3d = 65 => a + d = 65 / 3 => a + d = 21.67 ---- (2)

3. Now, we have two equations (1) and (2). We can solve these two equations to find the values of 'a' and 'd'.

   Subtract equation (2) from equation (1), we get:

   2a + 50d - (a + d) = 129 - 21.67 => a + 49d = 107.33 => a = 107.33 - 49d ---- (3)

4. Substitute equation (3) into equation (2), we get:

   107.33 - 49d + d = 21.67 => 107.33 - 48d = 21.67 => 48d = 107.33 - 21.67 => 48d = 85.66 => d = 85.66 / 48 => d = 1.78

5. Substitute d = 1.78 into equation (2), we get:

   a + 1.78 = 21.67 => a = 21.67 - 1.78 => a = 19.89

So, the first term 'a' is 19.89 and the common difference 'd' is 1.78.