Suppose we invest $\$50$ at time 0.5 and $\$200$ at time 1.5; the accumulation function that applies on the interval $[0,1]$ is $a(t) = 1+t^2$, and simple interest with $i = 5%$ applies on the interval $[1,2]$. Find the total accumulated value of the investments at time 2.
My attempt:
I know that I need to discount $\$50$ with the first accumulation function at time $0$, that is: $$\frac{50}{1+(0.5)^2}$$
Then I accumulate it at time 1: $$\frac{50*(1+1^2)}{1+(0.5)^2}$$
My problem starts when i want to accumulate it with simple interest on the interval $[1,2]$: Do I just multiply it by: $1+0.05*1$ to have an accumulated value of the first investment at time 2 equals to:
$$\frac{50*(1+1^2)*(1+0.05*1)}{1+(0.5)^2}$$
Or is this approach wrong?
Also How do I accumulate the second investment?
Any hints or suggestions would be highly appreciated.