Question

question
write a recursive formula for (a_{n}), the (n^{th}) term of the sequence 5, -20, 80, -320,....
answer attempt 1 out of 2
(a_{1}=)
(a_{n}=)

Explanation:

Step1: Identify the first - term

The first term of the sequence \(5, - 20,80,-320,\cdots\) is \(5\), so \(a_1 = 5\).

Step2: Find the common ratio

To find the common ratio \(r\), divide the second term by the first term. \(\frac{a_2}{a_1}=\frac{-20}{5}=- 4\). Also, \(\frac{a_3}{a_2}=\frac{80}{-20}=-4\), \(\frac{a_4}{a_3}=\frac{-320}{80}=-4\). The common ratio \(r=-4\).
For a geometric sequence, the recursive formula is \(a_n=r\times a_{n - 1}\). Here \(r = - 4\), so \(a_n=-4\times a_{n - 1}\).

Answer:

\(a_1 = 5\)
\(a_n=-4\times a_{n - 1}\)