Geometric Sequences and Exponential Functions
Key Questions

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.

Answer:
Geometric sequences are the discrete version of exponential functions, which are continuous.
Explanation:
Geometric sequences are formed by choosing a starting value and generating each subsequent value by multiplying the previous value by some constant called the geometric ratio. If a formula is provided, terms of the sequence are calculated by substituting
#n=0,1,2,3,...# into the formula. Note how only whole numbers are used, because it doesn't make sense to have a "one and threequarterth" term. With an exponential function, the inputs can be any real number from negative infinity to positive infinity. 
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# 
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.
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