5) A person invested some amount today @ \( 1.8 \% \) per quarter for 10 years find his investment he receives Rs. 10 million. \begin{tabular}{|l|l|l|l|}\hline a) 8.356 million & b) 4.13 million & c) 4.899 million & d) None of these \\ \hline\end{tabular}

To find the amount of investment, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (initial investment). - \( r \) is the annual interest rate (in decimal form). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the time the money is invested for in years. Given: - Annual interest rate \( r = 1.8\% = 0.018 \) (in decimal form). - Compounded quarterly, so \( n = 4 \). - Time period \( t = 10 \) years. - Amount received after 10 years \( A = 10 \) million. We need to find the initial investment \( P \). Substitute the given values into the formula: \[ 10,000,000 = P \left(1 + \frac{0.018}{4}\right)^{4 \times 10} \] Now, we can solve for \( P \) to find the initial investment. Calculate the value by following steps: - step0: Calculate: \(10000000\left(1+\frac{0.018}{4}\right)^{4\times 10}\) - step1: Divide the terms: \(10000000\left(1+\frac{9}{2000}\right)^{4\times 10}\) - step2: Add the numbers: \(10000000\left(\frac{2009}{2000}\right)^{4\times 10}\) - step3: Multiply the numbers: \(10000000\left(\frac{2009}{2000}\right)^{40}\) - step4: Simplify: \(10000000\times \frac{2009^{40}}{2000^{40}}\) - step5: Rewrite the expression: \(2000\times 5000\times \frac{2009^{40}}{2000^{40}}\) - step6: Reduce the numbers: \(5000\times \frac{2009^{40}}{2000^{39}}\) - step7: Rewrite the expression: \(1000\times 5\times \frac{2009^{40}}{2000^{39}}\) - step8: Rewrite the expression: \(1000\times 5\times \frac{2009^{40}}{1000^{39}\times 2^{39}}\) - step9: Reduce the numbers: \(5\times \frac{2009^{40}}{1000^{38}\times 2^{39}}\) - step10: Rewrite the expression: \(5\times \frac{2009^{40}}{125^{38}\times 8^{38}\times 2^{39}}\) - step11: Rewrite the expression: \(5\times \frac{2009^{40}}{5^{114}\times 8^{38}\times 2^{39}}\) - step12: Reduce the numbers: \(\frac{2009^{40}}{5^{113}\times 8^{38}\times 2^{39}}\) - step13: Multiply the fractions: \(\frac{2009^{40}}{5^{113}\times 2^{153}}\) The amount of investment is approximately 1.196734 × 10^7, which is equal to 12,967,340. Therefore, the correct answer is: c) 4.899 million