How do you write a rule for the nth term of the geometric sequence given the two terms #a_3=5, a_6=5000#?

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Answer 1 · 2017-03-19T07:16:06

Rule for the #n^(th)# term of the geometric sequence is #a_n=10^(n-1)/20#

If #m^(th)# term of a geometric sequence is #a_m# and #n^(th)# term of the sequence is #a_n#, then the common ratio #r# is given by

#r=(a_n/a_m)^(1/((n-m)))#

Here, we have #a_3=5# and #a_6=5000#, hence

#r=(5000/5)^(1/((6-3)))=1000^(1/(3))=root(3)1000=10#

and as #n^(th)# term of a geometric sequence is given by

#a_n=a_1xxr^(n-1)# and hence as #a_3=5#

#a_1=a_n/r^(n-1)=5/10^2=0.05#

and rule for the #n^(th)# term of the geometric sequence is

#a_n=0.05xx10^(n-1)=10^(n-1)/20#