suppose you could make a single \lump sum\ deposit of $3976, in an investment that provides an annual…

Explanation:

Step1: Recall compound interest formula

The formula for compound interest when compounded quarterly is ( FV = P\left(1 + \frac{r}{n}\right)^{nt} ), where ( P ) is the principal amount, ( r ) is the annual interest rate (in decimal), ( n ) is the number of times compounded per year, and ( t ) is the number of years. Here, ( P = 3976 ), ( r = 0.02 ) (since 2% = 0.02), ( n = 4 ) (compounded quarterly), and ( t = 34 ).

Step2: Calculate the exponent ( nt )

( nt=4\times34 = 136 )

Step3: Calculate the rate per period ( \frac{r}{n} )

( \frac{r}{n}=\frac{0.02}{4}=0.005 )

Step4: Calculate ( \left(1 + \frac{r}{n}\right)^{nt} )

( \left(1 + 0.005\right)^{136} \approx e^{0.005\times136}) (using the approximation ( a^b\approx e^{b\ln a} ) or directly calculating ( 1.005^{136} )). ( 1.005^{136}\approx1.9799 ) (using a calculator: ( 1.005^{136}=\text{exp}(136\times\ln(1.005))\approx\text{exp}(136\times0.0049875)\approx\text{exp}(0.6783)\approx1.9799 ))

Step5: Calculate the Future Value ( FV )

( FV = 3976\times1.9799\approx3976\times1.98 ) (approximating 1.9799 as 1.98 for easier calculation, but more accurately) ( 3976\times1.9799 = 3976\times(2 - 0.0201)=3976\times2-3976\times0.0201 = 7952- 80.0176=7871.9824) Using a more precise calculation of ( 1.005^{136} ): ( 1.005^{136}=e^{136\times\ln(1.005)}\approx e^{136\times0.00498752124}\approx e^{0.678302889}\approx1.97993 ) Then ( FV = 3976\times1.97993\approx3976\times1.97993 = 3976\times(2 - 0.02007)=7952-3976\times0.02007 = 7952 - 80.00832 = 7871.99168\approx7872.00 )

Answer:

($7872.00) (or more precisely, approximately ($7872))