SOLUTION: What is a recursive and an explicit formula for the sequence: 2/3, 4/9, 16/81 ...

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Question 614717: What is a recursive and an explicit formula for the sequence: 2/3, 4/9, 16/81 ...
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Wr'll find the explicit formula first:

Sequence 1 = 2%2F3, 4%2F9, 16%2F81, ...

That's the sequence

a1=%282%2F3%29%5E1, a2=%282%2F3%29%5E2, a3=%282%2F3%29%5E4, ...

The exponents follow the sequence

Sequence 2:  1, 2, 4, ...

That's the sequence:

20, 21, 22, ...

Those exponents follow the sequence:

Sequence 3:  0, 1, 2

which follow the sequence

1-1, 2-1, 3-1

which has nth term:

n-1

Therefore sequence 2 has nth term:

2n-1

Therefore sequence 1 has sequence:

an = %282%2F3%29%5E%282%5E%28n-1%29%29 

That's the explicit formula.

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So the sequence is

%282%2F3%29%5E%282%5E0%29, %282%2F3%29%5E%282%5E1%29, %282%2F3%29%5E%282%5E2%29, ...

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To find a recursion formula:

Divide the (n+1)st term by the nth term:

%282%2F3%29%5E%282%5E%28%28n%2B1%29-1%29%29÷%282%2F3%29%5E%282%5E%28n-1%29%29

%282%2F3%29%5E%282%5E%28n%29%29÷%282%2F3%29%5E%282%5E%28n-1%29%29

%282%2F3%29%5E%282%5En-2%5E%28n-1%29%29

%282%2F3%29%5E%282%5E%28n-1%29%282-1%29%29

%282%2F3%29%5E%282%5E%28n-1%29%281%29%29

%282%2F3%29%5E%282%5E%28n-1%29%29  

an+1 = an·(2%2F3)2n-1

And the complete recursion formula is:

a1 = 2%2F3, an+1 = an+1 = an·(2%2F3)2n-1

Edwin