The sequence 3, 12, 48, 192 appears to be increasing rapidly. To find an explicit rule for the nth term, we first look for a pattern in the ratios of consecutive terms.
If we divide each term by its predecessor, we get:
- 12 ÷ 3 = 4
- 48 ÷ 12 = 4
- 192 ÷ 48 = 4
This shows that each term is obtained by multiplying the previous term by 4. Thus, it is clear that the sequence is generated by a common ratio of 4. The explicit rule for the nth term can therefore be expressed as:
an = 3 × 4(n-1)
Where:
- an is the nth term of the sequence.
- n is the position of the term in the sequence (starting from n = 1).
Let’s confirm this rule by calculating the first few terms:
- For n = 1: a1 = 3 × 4(1-1) = 3 × 1 = 3
- For n = 2: a2 = 3 × 4(2-1) = 3 × 4 = 12
- For n = 3: a3 = 3 × 4(3-1) = 3 × 16 = 48
- For n = 4: a4 = 3 × 4(4-1) = 3 × 64 = 192
As expected, our calculated terms match the original sequence. Thus, the explicit rule for the nth term of the sequence is confirmed: an = 3 × 4(n-1).