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.Choices:- 64, 9/8 32, 8/9 187/9, 8/9 72, 9/8

Sure, let's solve this step by step.

1. We know that the sum of an arithmetic series is given by the formula: S = n/2 * (a + l) where n is the number of terms, a is the first term, and l is the last term.

2. We are given that the sum of the first three terms is 65. So, we can write the equation: 3/2 * (a + a + 2d) = 65 where d is the common difference. Simplifying this, we get 2a + 3d = 43.

3. We are also given that the sum of the middle three terms is 129. The middle term of an arithmetic series with an odd number of terms is also the average of the series. So, the middle three terms are (a + 24d), (a + 25d), and (a + 26d). We can write the equation: 3/2 * (3a + 75d) = 129. Simplifying this, we get 2a + 50d = 86.

4. Now we have a system of two equations: 2a + 3d = 43 and 2a + 50d = 86. We can solve this system to find the values of a and d.

5. Subtracting the first equation from the second, we get 47d = 43. So, d = 43/47 = 8/9.

6. Substituting d = 8/9 into the first equation, we get 2a + 3*(8/9) = 43. Solving for a, we get a = (43 - 24/9) / 2 = 187/9.

So, the first term of the series is 187/9 and the common difference is 8/9. Therefore, the correct choice is 187/9, 8/9.