answer:**Analysis:** This problem tests our understanding of the formula for the sum of a geometric series, the arithmetic of complex numbers, and their periodicity. It is a basic question. By using the formula for the sum of a geometric series and the arithmetic of complex numbers, we can derive the solution as follows. **Step-by-step solution:** 1. First, recognize that the given series is a geometric series with a common ratio of i. 2. Apply the formula for the sum of a geometric series, which is S_n = frac{a(1 - r^n)}{1 - r}, where a is the first term and r is the common ratio. In this case, a = i and r = i. 3. Substitute these values into the formula to get S_{2018} = frac{i(1 - i^{2018})}{1 - i}. 4. Simplify the expression by noting that i^{2018} = (i^4)^{504} cdot i^2 = 1^{504} cdot (-1) = -1. 5. Thus, the sum becomes S_{2018} = frac{i(1 - (-1))}{1 - i} = frac{2i}{1 - i}. 6. To eliminate the imaginary number in the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, which is 1 + i. This gives us S_{2018} = frac{2i(1 + i)}{(1 - i)(1 + i)} = frac{2i(1 + i)}{1^2 + 1^2} = frac{2i(1 + i)}{2} = i(1 + i) = -1 + i. Therefore, the sum of the series is boxed{-1 + i}.

answer:Let's denote the length of the shorter piece as S and the length of the longer piece as L. According to the problem, we have: S = 35 cm (given) The longer piece is 15 cm longer than twice the length of the shorter piece, so we can write: L = 2S + 15 Since we know the length of the shorter piece (S), we can substitute it into the equation for L: L = 2(35) + 15 L = 70 + 15 L = 85 cm Now, to find the length of the original board, we need to add the lengths of the shorter and longer pieces together: Original board length = S + L Original board length = 35 cm + 85 cm Original board length = 120 cm Therefore, the original board was boxed{120} cm long.

answer:If the first house cost 200,000 and the second house was twice as expensive, then the second house cost 200,000 x 2 = 400,000. To find the total price of both houses, we add the cost of the first house and the second house together: 200,000 (first house) + 400,000 (second house) = 600,000. Therefore, the total price of both houses was boxed{600,000} .

answer:Let's denote the principal amount that Peter invested as P. The simple interest formula is: [ text{Simple Interest} = P times r times t ] where ( P ) is the principal amount, ( r ) is the rate of interest per time period, and ( t ) is the time period. Given that the rate of interest is 50% or 0.5 per year, and the value of the bond grew to 100 at the end of 2 years, we can write the equation for the value of the bond at the end of 2 years as: [ P + (P times 0.5 times 2) = 100 ] Solving for ( P ): [ P + P times 1 = 100 ] [ 2P = 100 ] [ P = 50 ] Now, we know the principal amount ( P ) is 50. To find the value of the bond at the end of the total 6 years, we need to calculate the simple interest for the additional 4 years and add it to the principal amount. The simple interest for the additional 4 years is: [ text{Simple Interest for 4 years} = P times r times t ] [ text{Simple Interest for 4 years} = 50 times 0.5 times 4 ] [ text{Simple Interest for 4 years} = 50 times 2 ] [ text{Simple Interest for 4 years} = 100 ] Therefore, the value of the bond at the end of the total 6 years is: [ text{Value after 6 years} = P + text{Simple Interest for 4 years} ] [ text{Value after 6 years} = 50 + 100 ] [ text{Value after 6 years} = 150 ] The value of the bond at the end of the total 6 years is boxed{150} .