$[1]$ The yield to maturity for similar bonds is $2\%.$ Assume the yield remains at this level.

$[$a$]$ How much did you pay for the bond at time zero, $($price of the bond at time $0)?$

$[$b$]$ What is the price of the bond at time $1,$ after the first coupon has been paid?

$[$c$]$ What is the price at time $2?$

$[$d$]$ is the price increasing or decreasing over time?

$[2]$ Now, assume that the YTM is $10\%.$ Compute the prices at time $0,1,$ and $2.$ Is the price increasing or decreasing over time?

$[3]$ Compare $[1]$ and $[2].$ Is this what you expected? Why is the price falling in one of the cases, and increasing in the other? Is this going to be true for other bonds? Can you provide either a finance intuition or a math proof to support your claim?