We are given
sequence is 2 , 4 , 8 ,16 , 32 , .....
Firstly , we will check whether it is geometric sequence
Checking geometric sequence:
We will find common ratio between successive terms
and then we check whether they are equal
r1=(second term)/(first term)
[tex] r_1=\frac{4}{2}=2 [/tex]
r2=(third term)/(second term)
[tex] r_2=\frac{8}{4}=2 [/tex]
r3=(fourth term)/(third term)
[tex] r_3=\frac{16}{8}=2 [/tex]
r4=(fifth term)/(fourth term)
[tex] r_4=\frac{32}{16}=2 [/tex]
we can see that all four ratios are same
[tex] r_1=r_2=r_3=r_4=2 [/tex]
so, this is geometric sequence
Calculation of general term:
We got
common ratio is
[tex] r=2 [/tex]
Let's assume
number of terms is n
first term is 2
[tex] a_1=2 [/tex]
now, we can use formula
[tex] a_n=a_1 (r)^{n-1} [/tex]
we can plug values
and we get
[tex] a_n=2(2)^{n-1} [/tex]
[tex] a_n=2^1(2)^{n-1} [/tex]
[tex] a_n=(2)^{n-1+1} [/tex]
[tex] a_n=(2)^{n} [/tex]..................Answer
