We have: #a_(4) = 351# and #a_(7) = 13#
The #n#th term of a geometric sequence is given by:
#a_(n) = a_(1) r^(n - 1)#
Let's express the #4#th and #7#th terms using this rule:
#Rightarrow a_(4) = 351#
#Rightarrow a_(1) r^(4 - 1) = 351#
#Rightarrow a_(1) r^(3) = 351# ----------- #(i)#
and
#Rightarrow a_(7) = 13#
#Rightarrow a_(1) r^(7 - 1) = 13#
#Rightarrow a_(1) r^(6) = 13# ------------ #(ii)#
Then, let's divide #(ii)# by #(i)#:
#Rightarrow frac(a_(1) r^(6))(a_(1) r^(3)) = frac(13)(351)#
#Rightarrow r^(3) = frac(1)(27)#
#Rightarrow r = frac(1)(3)#
Now, let's find the first term by substituting this value for the common ratio into #(ii)#:
#Rightarrow a_(1) (frac(1)(3))^(6) = 13#
#Rightarrow frac(a_(1))(729) = 13#
#Rightarrow a_(1) = 9477#
Finally, let's substitute these values back into the rule for the #n#th term:
#therefore a_(n) = 9477 cdot (frac(1)(3))^(n - 1)#