Question

(b) classify the sequence and write its ( n^{\text{th}} ) term.

4, 12, 36, ...

the sequence is select .

( a_n = square )

Explanation:

Step1: Identify the type of sequence

To determine if the sequence is arithmetic or geometric, we check the ratios between consecutive terms.
For the sequence \(4, 12, 36, \dots\):

• The ratio between the second and first term: \(\frac{12}{4} = 3\)
• The ratio between the third and second term: \(\frac{36}{12} = 3\)

Since the ratio between consecutive terms is constant (\(r = 3\)), the sequence is a geometric sequence.

Step2: Find the \(n^{\text{th}}\) term of the geometric sequence

The formula for the \(n^{\text{th}}\) term of a geometric sequence is \(a_n=a_1\times r^{n - 1}\), where \(a_1\) is the first term and \(r\) is the common ratio.
Here, \(a_1 = 4\) and \(r=3\). So, \(a_n=4\times3^{n - 1}\)

Answer:

The sequence is a geometric sequence. \(a_n = 4\times3^{n - 1}\)