One of my actuarial exam problems asks to find $\ddot{s}_{\bar{2}|}$, given $\delta_t=\frac{2t}{10+t^2}$ $(t\geq 0)$.
Here's what I've done: From the definition of $\delta_t=\frac{A'(t)}{A(t)}$, $A(t)=10+t^2$ and it gives $A(0)=10$, so I found $a(t)=\frac{A(t)}{A(0)}=1+\frac{t^2}{10}$.
Now I'm struggling to understand what $\ddot{s}_{\bar{2}|}$ means, because I think it is a future value of annually invested money at Jan 1st, so $\ddot{s}_{\bar{2}|}=a(0)+a(1)+a(2)=1+(1+1/10)+(1+4/10)=3.5$, but it gives a wrong answer. (the correct one is $2.67$, I don't know why)
Since it's not a compound interest, I can't convert it to $a_{\bar{n}|}$...
Any help or hint would be appreciated.
I suppose $\ddot{s}_{\overline{2}|}$ means that there are two payments (at $t =0$ and $t=1$) of $1$ dollar each, and we want the value at the time after the final payment (i.e. at $t=2$). Also, I believe that $1$ dollar at time $k$ is worth $a(2)/a(k)$ dollars at time $2$. Thus the desired future value should be $\color{blue}{a(2)/a(0) + a(2)/a(1)}$. Using your formula for $a(t)$ will give $2.67$ (to the nearest cent).