Final answer:
The interest rate j2 for Jeff's couch purchase can be estimated using linear interpolation by calculating the present value of the semi-annual payments at two different interest rates and determining the nominal semi-annual interest rate between these bounds.
Step-by-step explanation:
To determine the interest rate j2 that Jeff is being charged for the couch, we need to compare the cash price with the total cost of the semi-annual payments. Linear interpolation is a method used to estimate unknown values that fall between two known values. However, before we can perform linear interpolation, we need to find two nominal interest rates (j2) that bound the actual rate by calculating the present value of the payments at different interest rates and comparing them to the cash price. Once we have these bounds, we can apply linear interpolation to estimate the nominal semi-annual interest rate j2. This approach is similar to determining the present value of a stream of payments, such as the interest and principal paid on a bond, as in the example with the $3000 bond at 8% interest. For Jeff's couch purchase, we would begin by assigning a lower bound interest rate (perhaps 0%) and calculating the present value of the payments. If this value is more than $1400, we would choose a higher interest rate as the upper bound and repeat the process. Once we have a present value below and above $1400 at different interest rates, we know the true interest rate is between these two bounds. Using linear interpolation, we can estimate the nominal interest rate that equates the present value of the annuity to the cash price of $1400.