6+12+18+24+…+6x = (0.0625)-84, what is the value of x?

Step-by-Step Solution

1. Simplify the Left-Hand Side (LHS): The expression is an arithmetic series where each term is a multiple of 6: 6 * (1 + 2 + 3 + ... + x). The sum of the first x natural numbers is (x * (x + 1)) / 2. Therefore, the LHS = 6 * (x * (x + 1)) / 2 = 3 * x * (x + 1) = 3x^2 + 3x.

2. Evaluate the Right-Hand Side (RHS): The RHS is (0.0625)^(-84). Note that 0.0625 can be written as 625 / 10000, which simplifies to 1 / 16. Since 1 / 16 is 1 / 2^4, or 2^(-4), the expression becomes (2^(-4))^(-84). Using exponent rules, this is 2^(336).

3. Solve for x: Given the options provided and the structure of the problem, there appears to be a discrepancy between the complex exponent on the RHS and the small integer options. Assuming the equation is intended to be solved using the options provided:

   • If x = 7: LHS = 3 * 7 * (7 + 1) = 3 * 7 * 8 = 168.

   • If x = 6: LHS = 3 * 6 * (6 + 1) = 3 * 6 * 7 = 126.

   • If x = 9: LHS = 3 * 9 * (9 + 1) = 3 * 9 * 10 = 270.

   • If x = 12: LHS = 3 * 12 * (12 + 1) = 3 * 12 * 13 = 468.

   Based on the provided correct answer of 7, the equation relies on the series sum equaling 168.