4.3 Summing a geometric series

Another sort of question.

How many phone calls in total made in the 5 phases?

$50000 + 50000 \times 0.95 + 50000\times 0.95^2 + \cdots + 500000 \times 0.95^4$ .

Add then up!

Or use the formula for the sum of a geometric series:

$$\fbox{$ P_0 + P_0 \theta + P_0 \theta^2 + \cdots + P_0 \theta^t = P_0 \frac{(1 - \theta^{t + 1})}{1 - \theta}.$ }$$

In our case, P₀ = 50000, $\theta = 0.95$ and t = 4.

$$\fbox{$50000 \frac{(1 - 0.95^{4 + 1})}{1 - 0.95} = 226219$ .}$$

Language:

The series is: $P_0, P_0 \theta, P_0 \theta^2, \cdots,$.

The t-th term is $P_0 \theta^{t-1}$.

The sum is $P_0 + P_0 \theta + P_0 \theta^2 + \cdots$.

Summary:

2 types of question.

Which term of a geometric series reaches a specified value (how long).

What is the sum of the first t + 1 terms of a geometric series (how many).