Question

question
write a recursive formula for $a_n$, the $n^{th}$ term of the sequence 3, - 12, 48, - 192, ....
answer attempt 1 out of 2
$a_1=square$
$a_n=square$
submit answer $a_{n - 1}$

Explanation:

Step1: Identify the first - term

The first term of the sequence \(3, - 12,48,-192,\cdots\) is \(3\), so \(a_1 = 3\).

Step2: Find the common ratio

To find the common ratio \(r\), divide the second term by the first term. \(\frac{a_2}{a_1}=\frac{-12}{3}=- 4\). Check with other consecutive terms: \(\frac{a_3}{a_2}=\frac{48}{-12}=-4\), \(\frac{a_4}{a_3}=\frac{-192}{48}=-4\). The common ratio \(r=-4\).

Step3: Write the recursive formula

For a geometric sequence, the recursive formula is \(a_n=r\times a_{n - 1}\). Since \(r = - 4\), the recursive formula is \(a_n=-4\times a_{n - 1}\).

Answer:

\(a_1 = 3\)
\(a_n=-4\times a_{n - 1}\)