I've been given the following expressions for regular payments with regular annual compounding (Annuities):
(1) $$ F - (1 + r)F $$ (2) $$ F = d \sum^T _{t=1} (1+r)^{T-t}$$
where: ($F$) is a Future Value, ($r$) is the Rate of Interest, ($d$) is the payment amount, ($T$) is the total count of payments and ($t$) is the payment number. I understand how to get to this point,
(3) $$ d\left[\sum^T_{t=1}(1+r)^{T-t} - \sum^T_{t=1}(1+r)^{T-t+1}\right] $$
but somehow my tutor has managed to manipulate the expression to,
(4) $$ -d[(1+r)^T - 1] $$
Does anyone know what my tutor did to go from (3) to (4)?
It's just a telescoping property. The easiest way to understand it is to pick a value for $T$, say $T = 3$, and write out the sums explicitly:
$$\sum_{t=1}^T (1+r)^{T-t} = (1+r)^2 + (1+r)^1 + (1+r)^0,$$ and $$\sum_{t=1}^T (1+r)^{T-t+1} = (1+r)^3 + (1+r)^2 + (1+r)^1.$$ So their difference is simply $$-(1+r)^3 + (1+r)^0 = -((1+r)^3 - 1),$$ because all the intermediate terms cancel. Now generalize this for any positive integer $T$, and you can see where the result comes from.