GEOMETRIC PROGRESSION  
Analysis  
6. SUM OF n TERMS OF A GEOMETRIC PROGRESSION  
We are going to try and find a formula which allows us to work out the sum of n terms in a geometric progression. 

10. Let's imagine we want
to find the sum of the first ten terms:
If we multiply the terms we want to add together in the sequence by the common ratio we get another sequence which is almost the same. If we increase step_1 (1, 2, ...) we can see that the terms in the resulting sequence are nearly the same. Try another number of terms to be added (11, 12, ..., 100, ...) and check that this is still the case. If we subtract one sum from the other we eliminate all the identical terms, as you can see in step_2 (1, 2, ...). The general formula can be seen in step_3 (1, 2, 3). 

7. SUM OF ALL TERMS WHEN r <1  
When the common ratio of a geometric progression is a number between 1 and 1 an infinite number of terms can be added together (the sum to infinity), as we can see in this window. 

11. Look at the sum of the first five terms. Increase the number of terms which are added together and note that their sum gets closer and closer to a certain number. Change the first term and the common ratio and try other progressions. Find the expression, based on the formula used in the previous section, which allows you to find the sum of all the terms, bearing in mind that the last term could be considered virtually zero. The general formula can be seen in step_1. 
Juan Madrigal Muga  
Spanish Ministry of Education. Year 2002  
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