Question

Write the explicit formula for the geometric sequence 2, 8, 32, 128

Answer 1

The first term, a, is 2.  The common ratio, r, is 4.  Thus,

a_(n+1) = 2(4)^(n).

Check:  What's the first term?  Let n=1.  Then we get 2(4)^1, or 8.  Is that correct?  No.

Try this instead:

a_(n) = a_0*4^(n-1).  Is this correct?    Seeking the first term (n=1), does this formula produce 2?      2*4^0 = 2*1 = 2.  YES.

The desired explicit formula is   a_(n) = a_0*4^(n-1), where n begins at 1.