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1.Kathryn deposits 100 into an account at the beginning of each 4-year period for 40 years. The account credits interest at an annual effective interest rate of i. The accumulated amount in the account at the end of 40 years is X, which is 5 times the accumulated amount in the account at the end of 20 years. Calculate X. Step-by-step explanation: Using the formula for calculating accumulated annuity amount F = P × ([1 + I]^N - 1 )/I Where P is the payment amount. I is equal to the interest (discount) rate and N number of duration For 40 years, X = 100[(1 + i)^40 + (1 + i)^36 + · · ·+ (1 + i)^4] =[100 × (1+i)^4 × (1 - (1 + i)^40]/1 − (1 + i)^4 For 20 years, Y = A(20) = 100[(1+i)^20+(1+i)^16+· · ·+(1+i)^4] Using X = 5Y (5 times the accumulated amount in the account at the ned of 20 years) and using a difference of squares on the left side gives 1 + (1 + i)^20 = 5 so (1 + i)^20 = 4 so (1 + i)^4 = 4^0.2 = 1.319508 Hence X = [100 × (1 + i)^4 × (1 − (1 + i)^40)] / 1 − (1 + i)^4 = [100×1.3195×(1−4^2)] / 1−1.3195 X = 6194.84 2. A perpetuity costs 77.1 and makes annual payments at the end of the year. The perpetuity pays 1 at the end of year 2, 2 at the end of year 3, .... , n at the end of year (n+1). After year (n+1), the payments remain constant at n. The annual effective interest rate is 10.5%. Calculate n 3.1000 is deposited into Fund X, which earns an annual effective rate of 6%. At the end of each year, the interest earned plus an additional100 is with ...
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