Calculating the Net Present Value (NPV)

Question

When input the apropriate data is subbed into the equation we get:

$$NPV=\sum_{t=0}^\infty\frac{200}{1.1^t}$$

I have been told that the second term looks like $$\sum_{t=0}^\infty\frac{200}{1.1^t} =\frac{200}{0.1}=2000$$

Could someone explain how the second term can end up being written as $\frac{200}{0.1}$?

Answer 1

Hint :

$\frac {200} {1.1^i}$ is the general term of a geometric sequence.

Answer 2

When you sum a geometric series, you have $$ S_N = \sum_{k=0}^N a^k = 1 + a + a^2 + \ldots + a^N\\ aS_n = a\sum_{k=0}^N a^k = a + a^2 + a^3 + \ldots + a^{N+1} $$ now subtract both sides to get $$ S_N - aS_N = 1 - a^{N+1}, $$ which implies $$ S_N = \frac{1-a^{N+1}}{1-a}. $$

As $N \to \infty$, if $0 < a < 1$, we have $a^{N+1} \to 0$, so $$ \lim_{N \to \infty} S_N = \frac{1}{1-a}. $$

Can you simplify your series to fit this result?