Amount of interest Find the amount of interest earned between time t and n where $t<n$; if $I_r=r$ for some positive integer r. Answer is $\frac{1}{2}(n^2+n-t^2-t)$
$I_{[t,n]}=A(n)-A(t)$
$I_{[0,r]}=A(r)-A(0)=r$
$A(r)=A(O)+r$
For $t<r<n$
$I_{[r,n]}=A(n)-A(r)$ ;(1)
$I_{[t,r]}=A(r)-A(t)$ ;(2)
Cannot grasp any further concept.

$$
A(n)-A(t)=\sum_{r=t+1}^n I_{[0,r] }=\sum_{r=t+1}^n r = \sum_{r=1}^n r-\sum_{r=1}^t r=\frac{n(n + 1)}{2}-\frac{t(t + 1)}{2}=\frac{1}{2}(n^2 + n- t^2 - t)
$$
