Question

Determine whether the sequence converges or diverges. if it converges give the limit.

48, 12, 3, 3/4, ...

Answer 1

The sequence converges.
The limit of the sequence is 0.

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This is a geometric sequence with the first term a = 48 and r = 1/4 = 0.25 as the common ratio. Since |r| < 1, this means that the terms are steadily getting smaller and smaller. The terms will approach 0 but not actually get there.

Note: if you added up all the terms of this infinite sequence, then the sum would be S = a/(1-r) = 48/(1-0.25) = 64