Alternative way to calculate sum of geometric series

I have been asked to find $S_8$ of a geometric series in which $U_3$ is 32 and $U_6$ is 4.

I have done this by.

Step 1) Drawing a chart to find the first term and the common difference.

Step 2) Using $S_n = a*\frac{1-r^n}{1-r}$ to calculate the sum

It is seen that $S_8 = 128*\frac{1-(1/2)^n}{1-(1/2)} = 255$

However this could have been done by simply adding the values on the chart.

I want to find out if there is another way to do it that only uses formulas? as this seems a bit easy for my university engineering maths exam.

Answer

As you pointed out you can just add the values up in the chart to get the value of the series. Hence by the definition of a series as repeated algorithmic addition. A nice result of developing the frame work of series is the geometric series, because it has a nice formula equivalent to adding up the first n-terms of your series (as you pointed out in step 2).

In other words the easier way of adding up the first n-terms of a geometric series is the formula you pointed out in part 2, and the long way of going about it is to add up the first n-terms individually. I know it may seem like adding up the first 8 terms for this case is the easy method, but if you were asked to add up the first 2000 terms, then the formula is by far superior.

Another nice relationship is that the ratio of any consecutive terms is r, and you could use this fact to find r if you are given that the series is geometric.

In your example $U6/U3 = r^3$.