Question

question write a recursive formula for (a_{n}), the (n^{th}) term of the sequence 2, -6, 18, -54, .... answer attempt 1 out of 2 (a_{1}=) (a_{n}=) submit answer (a_{n - 1})

Explanation:

Step1: Identify first - term

The first term of the sequence \(2, - 6,18,-54,\cdots\) is \(2\), so \(a_1 = 2\).

Step2: Find common ratio

To find the relationship between consecutive terms, divide the second - term by the first - term: \(\frac{-6}{2}=-3\), divide the third - term by the second - term: \(\frac{18}{-6}=-3\), divide the fourth - term by the third - term: \(\frac{-54}{18}=-3\). The common ratio \(r=-3\).
For a geometric sequence, the recursive formula is \(a_n = r\times a_{n - 1}\). Here, \(a_n=-3\times a_{n - 1}\) for \(n\geq2\).

Answer:

\(a_1 = 2\)
\(a_n=-3\times a_{n - 1},n\geq2\)