Question

What is the explicit rule for this geometric sequence?

2/9,2/3,2,6,...

a) an=3(29)ⁿ⁻¹

b) an=3(29)ⁿ

c) an=29⋅3ⁿ⁻¹

d) an=29⋅3ⁿ

Answer 1

To find the explicit rule for a geometric sequence, we need to identify the common ratio (\(r\)). In this case, the common ratio is \(\frac{2}{9} \div \frac{2}{3} = \frac{2}{9} \times \frac{3}{2} = \frac{1}{3}\).

The explicit rule for a geometric sequence is given by \(a_n = a_1 \times r^{(n-1)}\), where \(a_1\) is the first term and \(r\) is the common ratio.

So, for the given sequence, the explicit rule is:

\[a_n = \frac{2}{9} \times \left(\frac{1}{3}\right)^{n-1}\]

Which can be simplified as:

\[a_n = \frac{2}{3^n}\]

So, the correct option is:

c) \(a_n = \frac{2}{3^n}\)