I'm currently studying a finance textbook and they just give me the transformation from the sum to the simplification.
They say it's a geometric progression but they don't show the demonstration.
Could you please give me all the steps with all the details to go from from $$ Present Value = 26.4 \times \left[ \frac{1}{(1+10\%)^1} + \frac{1}{(1+10\%)^2} + ... + \frac{1}{(1+10\%)^5} \right] =100 $$
to
$$ Present Value = 26.4 \times \frac{1-(1+10\%)^{-5}}{10\%} =100 $$ ?
Remember that $$ \frac{1}{(1+10\%)^{n}} = \left(\frac{1}{1+10\%}\right)^{n} $$
Suppose we have $$a = \frac{1}{1+10\%}=\frac{1}{1.1}$$ and $$S=a+a^2+a^3+a^4+a^5$$ Then multiply by $a$ $$a\times S=a^2+a^3+a^4+a^5+a^6$$ Subtract the second from the first to obtain $$ 1\times S - a\times S = (1-a)\times S =a-a^6 =a \times 1 -a\times a^5 =a\times (1-a^5)$$ (the point is that most terms simply cancel) so that $$S=\frac a{1-a}(1-a^5)$$
Then you put $ a=\frac 1{1.1} $
Finally $$ S=\frac{1}{10\%}\times (1-(1+10\%)^{5^{-1}})) \\ S=\frac{1-(1+10\%)^{-5}}{10\%} $$
More generally, for an interest rate $r$, $a=\frac 1{1+r}$, whence $$S=\frac 1r\left(1-\frac 1{(1+r)^5}\right)$$