If \( \$ 18,000 \) is invested at \( 2.5 \% \) for 30 years, find the future value if the interest is compounded the following ways. (Round your answers to the nearest cent.) (a) annually \$ 37756.22 Very nice! (b) semiannually \$ \( \qquad \) (c) quarterly \$ \( \qquad \) (d) monthly \$ \( \qquad \) (e) daily ( \( N=360 \) ) \$ \( \qquad \) \( \times \) (f) every minute ( \( N=525,600 \) ) \$ \( \qquad \) \( \times \) (g) continuously \( \$ 37756.22 \) That's itt (h) simple (not compounded) \$ \( \qquad \)

To find the future value of an investment compounded in different ways, we can use the formula for compound interest: \[ FV = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( FV \) = future value - \( P \) = principal amount (initial investment) - \( r \) = annual interest rate (as a decimal) - \( n \) = number of times interest is compounded per year - \( t \) = number of years the money is invested For the given problem: - \( P = 18000 \) - \( r = 0.025 \) - \( t = 30 \) Now, we will calculate the future value for each compounding frequency. ### (b) Semiannually For semiannual compounding, \( n = 2 \). \[ FV = 18000 \left(1 + \frac{0.025}{2}\right)^{2 \times 30} \] ### (c) Quarterly For quarterly compounding, \( n = 4 \). \[ FV = 18000 \left(1 + \frac{0.025}{4}\right)^{4 \times 30} \] ### (d) Monthly For monthly compounding, \( n = 12 \). \[ FV = 18000 \left(1 + \frac{0.025}{12}\right)^{12 \times 30} \] ### (e) Daily ( \( N=360 \) ) For daily compounding with \( N = 360 \), \( n = 360 \). \[ FV = 18000 \left(1 + \frac{0.025}{360}\right)^{360 \times 30} \] ### (f) Every minute ( \( N=525,600 \) ) For compounding every minute, \( n = 525600 \). \[ FV = 18000 \left(1 + \frac{0.025}{525600}\right)^{525600 \times 30} \] ### (h) Simple (not compounded) For simple interest, the formula is: \[ FV = P(1 + rt) \] Now, let's calculate the future values for (b), (c), (d), (e), (f), and (h). I'll perform these calculations now. Calculate the value by following steps: - step0: Calculate: \(18000\left(1+\frac{0.025}{2}\right)^{2\times 30}\) - step1: Divide the terms: \(18000\left(1+\frac{1}{80}\right)^{2\times 30}\) - step2: Add the numbers: \(18000\left(\frac{81}{80}\right)^{2\times 30}\) - step3: Multiply the numbers: \(18000\left(\frac{81}{80}\right)^{60}\) - step4: Simplify: \(18000\times \frac{81^{60}}{80^{60}}\) - step5: Rewrite the expression: \(80\times 225\times \frac{81^{60}}{80^{60}}\) - step6: Reduce the numbers: \(225\times \frac{81^{60}}{80^{59}}\) - step7: Rewrite the expression: \(25\times 9\times \frac{81^{60}}{80^{59}}\) - step8: Rewrite the expression: \(25\times 9\times \frac{81^{60}}{5^{59}\times 16^{59}}\) - step9: Rewrite the expression: \(5^{2}\times 9\times \frac{81^{60}}{5^{59}\times 16^{59}}\) - step10: Reduce the numbers: \(9\times \frac{81^{60}}{5^{57}\times 16^{59}}\) - step11: Multiply the fractions: \(\frac{9\times 81^{60}}{5^{57}\times 16^{59}}\) - step12: Multiply: \(\frac{9^{121}}{5^{57}\times 16^{59}}\) Calculate or simplify the expression \( 18000 * (1 + 0.025/12)^(12*30) \). Calculate the value by following steps: - step0: Calculate: \(18000\left(1+\frac{0.025}{12}\right)^{12\times 30}\) - step1: Divide the terms: \(18000\left(1+\frac{1}{480}\right)^{12\times 30}\) - step2: Add the numbers: \(18000\left(\frac{481}{480}\right)^{12\times 30}\) - step3: Multiply the numbers: \(18000\left(\frac{481}{480}\right)^{360}\) - step4: Simplify: \(18000\times \frac{481^{360}}{480^{360}}\) - step5: Rewrite the expression: \(240\times 75\times \frac{481^{360}}{480^{360}}\) - step6: Rewrite the expression: \(240\times 75\times \frac{481^{360}}{240^{360}\times 2^{360}}\) - step7: Reduce the numbers: \(75\times \frac{481^{360}}{240^{359}\times 2^{360}}\) - step8: Rewrite the expression: \(15\times 5\times \frac{481^{360}}{240^{359}\times 2^{360}}\) - step9: Rewrite the expression: \(15\times 5\times \frac{481^{360}}{15^{359}\times 16^{359}\times 2^{360}}\) - step10: Reduce the numbers: \(5\times \frac{481^{360}}{15^{358}\times 16^{359}\times 2^{360}}\) - step11: Rewrite the expression: \(5\times \frac{481^{360}}{5^{358}\times 3^{358}\times 16^{359}\times 2^{360}}\) - step12: Reduce the numbers: \(1\times \frac{481^{360}}{5^{357}\times 3^{358}\times 16^{359}\times 2^{360}}\) - step13: Multiply the fractions: \(\frac{481^{360}}{5^{357}\times 3^{358}\times 2^{1796}}\) Calculate or simplify the expression \( 18000 * (1 + 0.025*30) \). Calculate the value by following steps: - step0: Calculate: \(18000\left(1+0.025\times 30\right)\) - step1: Multiply the numbers: \(18000\left(1+0.75\right)\) - step2: Add the numbers: \(18000\times 1.75\) - step3: Multiply the numbers: \(31500\) Calculate or simplify the expression \( 18000 * (1 + 0.025/4)^(4*30) \). Calculate the value by following steps: - step0: Calculate: \(18000\left(1+\frac{0.025}{4}\right)^{4\times 30}\) - step1: Divide the terms: \(18000\left(1+\frac{1}{160}\right)^{4\times 30}\) - step2: Add the numbers: \(18000\left(\frac{161}{160}\right)^{4\times 30}\) - step3: Multiply the numbers: \(18000\left(\frac{161}{160}\right)^{120}\) - step4: Simplify: \(18000\times \frac{161^{120}}{160^{120}}\) - step5: Rewrite the expression: \(80\times 225\times \frac{161^{120}}{160^{120}}\) - step6: Rewrite the expression: \(80\times 225\times \frac{161^{120}}{80^{120}\times 2^{120}}\) - step7: Reduce the numbers: \(225\times \frac{161^{120}}{80^{119}\times 2^{120}}\) - step8: Rewrite the expression: \(25\times 9\times \frac{161^{120}}{80^{119}\times 2^{120}}\) - step9: Rewrite the expression: \(25\times 9\times \frac{161^{120}}{5^{119}\times 16^{119}\times 2^{120}}\) - step10: Rewrite the expression: \(5^{2}\times 9\times \frac{161^{120}}{5^{119}\times 16^{119}\times 2^{120}}\) - step11: Reduce the numbers: \(9\times \frac{161^{120}}{5^{117}\times 16^{119}\times 2^{120}}\) - step12: Multiply the fractions: \(\frac{9\times 161^{120}}{5^{117}\times 2^{596}}\) Calculate or simplify the expression \( 18000 * (1 + 0.025/360)^(360*30) \). Calculate the value by following steps: - step0: Calculate: \(18000\left(1+\frac{0.025}{360}\right)^{360\times 30}\) - step1: Divide the terms: \(18000\left(1+\frac{1}{14400}\right)^{360\times 30}\) - step2: Add the numbers: \(18000\left(\frac{14401}{14400}\right)^{360\times 30}\) - step3: Multiply the numbers: \(18000\left(\frac{14401}{14400}\right)^{10800}\) - step4: Simplify: \(18000\times \frac{14401^{10800}}{14400^{10800}}\) - step5: Rewrite the expression: \(3600\times 5\times \frac{14401^{10800}}{14400^{10800}}\) - step6: Rewrite the expression: \(3600\times 5\times \frac{14401^{10800}}{3600^{10800}\times 4^{10800}}\) - step7: Reduce the numbers: \(5\times \frac{14401^{10800}}{3600^{10799}\times 4^{10800}}\) - step8: Rewrite the expression: \(5\times \frac{14401^{10800}}{25^{10799}\times 144^{10799}\times 4^{10800}}\) - step9: Rewrite the expression: \(5\times \frac{14401^{10800}}{5^{21598}\times 144^{10799}\times 4^{10800}}\) - step10: Reduce the numbers: \(1\times \frac{14401^{10800}}{5^{21597}\times 144^{10799}\times 4^{10800}}\) - step11: Multiply the fractions: \(\frac{14401^{10800}}{5^{21597}\times 144^{10799}\times 4^{10800}}\) Calculate or simplify the expression \( 18000 * (1 + 0.025/525600)^(525600*30) \). Calculate the value by following steps: - step0: Calculate: \(18000\left(1+\frac{0.025}{525600}\right)^{525600\times 30}\) - step1: Divide the terms: \(18000\left(1+\frac{1}{21024000}\right)^{525600\times 30}\) - step2: Add the numbers: \(18000\left(\frac{21024001}{21024000}\right)^{525600\times 30}\) - step3: Multiply the numbers: \(18000\left(\frac{21024001}{21024000}\right)^{15768000}\) - step4: Simplify: \(18000\times \frac{21024001^{15768000}}{21024000^{15768000}}\) - step5: Rewrite the expression: \(18000\times \frac{21024001^{15768000}}{18000^{15768000}\times 1168^{15768000}}\) - step6: Reduce the numbers: \(1\times \frac{21024001^{15768000}}{18000^{15767999}\times 1168^{15768000}}\) - step7: Multiply the fractions: \(\frac{21024001^{15768000}}{18000^{15767999}\times 1168^{15768000}}\) Now, let's summarize the future values calculated for each compounding method: ### (b) Semiannually \[ FV \approx 37929.26 \] ### (c) Quarterly \[ FV \approx 38017.16 \] ### (d) Monthly \[ FV \approx 38076.28 \] ### (e) Daily ( \( N=360 \) ) \[ FV \approx 38105.01 \] ### (f) Every minute ( \( N=525,600 \) ) \[ FV \approx 38106.00 \] ### (h) Simple (not compounded) Using the simple interest formula: \[ FV = 18000 \times (1 + 0.025 \times 30) = 18000 \times (1 + 0.75) = 18000 \times 1.75 = 31500 \] ### Final Results - (b) Semiannually: \$ 37,929.26 - (c) Quarterly: \$ 38,017.16 - (d) Monthly: \$ 38,076.28 - (e) Daily: \$ 38,105.01 - (f) Every minute: \$ 38,106.00 - (h) Simple: \$ 31,500.00 If you need any further assistance or calculations, feel free to ask!