How do you write a rule for the nth term of the geometric sequence given the two terms #a_4=351, a_7=13#?

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Answer 1 · 2017-05-17T19:50:19

#a_(n) = 9477 cdot (frac(1)(3))^(n - 1)#

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)#