Compound interest as a geometric progression

Statement

A capital of $1000\ \text{€}$ is invested at $5\%$ compounded annually. Write the accumulated value after $n$ years as a term of a geometric progression and compute the value after $8$ years.

Solution

The yearly amounts form a geometric progression with first term $a_0 = 1000$ and ratio $q = 1{.}05$: $a_n = 1000\cdot 1{.}05^{\,n}$. $a_8 = 1000\cdot 1{.}05^{8} = 1000\cdot 1{.}477455 \approx \boxed{1477{.}46\ \text{€}}.$