The sum of the first four terms of an AP is 2/3 and the sum of the first eight term is -1.find the sum of the first twelve terms

S4 = 4/2 (2a+3d) = 2/3

S8 = 8/2 (2a+7d) = -1
a = 37/96 and d = -7/48
S12 = 12/2 (37/48 + 11(-7/48)) = -5

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Did you know that an Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant? In this case, the sum of the first four terms of the AP is 2/3, and the sum of the first eight terms is -1. To find the sum of the first twelve terms of the AP, we can use the formula for the sum of an AP. This formula states that the sum of 'n' terms in an AP is equal to n/2 times the sum of the first and last terms. By using this formula, we can calculate the sum of the first twelve terms by finding the sum of the first and last terms. The sum of the first four terms is 2/3, which means each term has a value of (2/3)/4 = 1/6. Similarly, the sum of the first eight terms is -1, indicating that each term has a value of (-1)/8 = -1/8. Using these values, we can determine that the first term is 1/6, and the twelfth term is -1/8. Plugging these values into the formula, we get the sum of the first twelve terms of the AP as 12/2 * (1/6 + -1/8). Therefore, the sum of the first twelve terms of this AP is 3/8.