Could someone please help me with these two math questions?

Question

1. What is the explicit rule for the geometric sequence? 120, 40, 40/3, 40/9, 40/27, ..

2. What is the recursive rule for the geometric sequence? 10, −80, 640, −5120, ...

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Thank you to anyone who helps!

Answer 1

1. What is the explicit rule for the geometric sequence? 120, 40, 40/3, 40/9, 40/27, ..

 

        a_n = 360/3ⁿ          [for n>0]

 

    Special note:  The Explicit Rule depends on the value of n, not on any other value in the sequence.

 

 

2. What is the recursive rule for the geometric sequence? 10, −80, 640, −5120, ...

 

      a₁ = 10

      a_n = -8a_n-1      [for n>1]

 

   Special note:  To find any term, a_n, you must recursively find all the previous terms (back to the base case).

 

Answer 2

You can always find the rule by dividing any element by the element before it in the series.  So, in the first series, 40 / 120 = 40/3 / 40 = ... = 1/3.  Every element is one third of the previous element.

 

The second series works the same way, except the sign alternates.  So -80 / 10 = -8, 640 / -80 = -8, etc.  Every element is equal to the previous element times -8.