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A mathematicsgeometric sequence is a sequence where each consecutive term is derived from the previous term multiplied by a fixed number called a common ratio.so the sequence like 1,5,25,125 and so on are the examples of a geometric sequence. The patter is that we are always multiplying the previous term by 5 .Not every sequence has pattern in multiplication is geometric, it is considered as geometric sequence only if it is been multiplied by the same number each and every time.
So the general formula for the geometric sequence is this: a of n is equal to a of 1 times r ^n-1, where a of 1 is the first term of the sequence and r is the common ratio. Let’s determine the common ratio of the geometric sequence using an example where a of n =5(3)^n ,The very first we need to find the first fourth term, so a of 1 = (5)(3)^1=15. Next step a of 2 = (5)(3)^2=45. Next will be a of 3 = (5) (3) ^3=135. And a of 4= (5)(3)^4=405. Here generally it’s good to write always a of 4 term of a sequence, just to confirm that the common ratio remains the same. And from the observation we can see that the common ration is going to be 3.
Let’s look out with one more example where we determine what would be the general formula for that. With the sequence given as 16,-4, 1,-1/4 and so on. We know the general formula of geometric sequence is of n =a of 1 times r ^n-1. Here we need to find a first term and r, the common ratio. It is obvious that the first term is going to be 16. So a of 1 =16. The sum of the n terms infinite terms of the sequence is S of n = a of 1 (1-r^n)/1-r where, r is not equal zero. And a 1 is our first term and r is our common ratio.
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