Final answer:
To find the value of a, take the twelfth root of 37. The value of a is approximately 1.742. To find S12, use the formula for the sum of an arithmetic series: S12 = (12/2)(2a + (n-1)d). Substitute the given values and simplify to find that S12 is equal to 218.904.
Step-by-step explanation:
The question states that a12 = 37 and d = 3, and asks to find the value of a and the sum of the first 12 terms (S12). To find a, we can take the twelfth root of 37, since a12 is equal to 37. Using a calculator, we find that a ≈ 1.742. To find S12, we can use the formula for the sum of an arithmetic series: S12 = (n/2)(2a + (n-1)d), where n is the number of terms, a is the first term, and d is the common difference. Substituting the values given, we have S12 = (12/2)(2(1.742) + (12-1)(3)), which simplifies to S12 = 6(3.484 + 33) = 6(36.484) = 218.904.