I am trying to prove an identity and quite not get there. The following is the premise.

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> One deposits $\$1$ at time $t=1,2, \cdots ,n$. evenly spaced. The effective interest per payment is $i$. Let $I\_t$ be the interest payable at time $t$. Show that $I\_t=(1+i)^{t-1}-1$.
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I am thinking that this is an annuity immediate, the accumulated value at time $t$ must be 

$$s\_{\overline t \rceil i}=\frac{(1+i)^t-1}{i}$$

Similarly that of time $t-1$ is

$$s\_{\overline {t-1 }\rceil i}=\frac{(1+i)^{t-1}-1}{i}$$

so since $I\_t$ can be simply figured out by

$$s\_{\overline {t}\rceil i}-s\_{\overline {t-1 }\rceil i} = (1+i)^{t-1}$$

Where is the correct answer getting its $-1$ from?

The only reason I could think of why I am not getting this would be because I am not really understanding the meaning of "interest payable at time $t$". Can someone help me out?