Geometric Sequences and Exponential Functions
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Key Questions

A geometric sequence is given by a starting number, and a common ratio.
Each number of the sequence is given by multipling the previous one for the common ratio.
Let's say that your starting point is
#2# , and the common ratio is#3# . This means that the first number of the sequence,#a_0# , is 2. The next one,#a_1# , will be#2 \times 3=6# . In general, we have that#a_n=3a_{n1}# .If the starting point is
#a# , and the ratio is#r# , we have that the generic element is given by#a_n=ar^n# . This means that we have several cases: If
#r=1# , the sequence is constantly equal to#a# ;  If
#r=1# , the sequence is alternatively equal to#a# and#a# ;  If
#r>1# , the sequence grows exponentially to infinity;  If
#r<1# , the sequence grows to infinity, assuming alternatively positive and negative values;  If
#1<r<1# , the sequence exponentially decreases to zero;  If
#r=0# , the sequence is constantly zero, from the second term on.
 If

It means that the next term can always be obtained by multiplying a current term by
#r# .#a_1=a# by multiplying by
#r# ,#a_2=ar# by multiplying by
#r# ,#a_3=ar^2#
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I hope that this was helpful.

Geometric sequences are exponential functions restricted to the natural numbers.
For example, consider
#f(x) = 2^x# . We know that the domain of that function is the set of real numbers, but if we restrict the domain to the natural numbers, we'll have#f(x) = 2^x# such that#x = 0, 1, 2, ...# forming the sequence#(1, 2, 4, 8, ...)# .Note that
#f(x) = 2^x# with the set of real numbers as the domain isn't a geometric sequence because, in sequences, there are no such thing as the#2,5655345th# term, only the 1st, 2nd, 3rd, and so on, terms. 
A geometric sequence is always of the form
#t_n=t_"n1"*r# Every next term is
#r# times as large as the one before.So starting with
#t_0# (the "start term") we get:
#t_1=r*t_0#
#t_2=r*t_1=r*r*t_0=r^2*t_0#
......
#t_n=r^n*t_0# Answer:
#t_n=r^n*t_0#
#t_0# being the start term,#r# being the ratioExtra:
If#r>1# then the sequence is said to be increasing
if#r=1# then all numbers in the sequence are the same
If#r<1# then the sequence is said to be decreasing ,
and a total sum may be calculated for an infinite sequence:
sum#sum=t_0/(1r)# Example :
The sequence#1,1/2,1/4,1/8...#
Here the#t_0=1# and the ratio#r=1/2#
Total sum of this infinite sequence:
#sum=t_0/(1r)=1/(11/2)=2#
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Exponents and Exponential Functions

1Exponential Properties Involving Products

2Exponential Properties Involving Quotients

3Negative Exponents

4Fractional Exponents

5Scientific Notation

6Scientific Notation with a Calculator

7Exponential Growth

8Exponential Decay

9Geometric Sequences and Exponential Functions

10Applications of Exponential Functions