Wr'll find the explicit formula first:

Sequence 1 = , , , ...

That's the sequence

a1=, a2=, a3=, ...

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 =  

That's the explicit formula.

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So the sequence is

, , , ...

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To find a recursion formula:

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

÷

÷







  

an+1 = an·()2n-1

And the complete recursion formula is:

a1 = , an+1 = an+1 = an·()2n-1

Edwin