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:

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

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