Question

find a general term definition for this sequence.\

$$\begin{cases}a_0 = 5\\\\a_n = 2a_{n - 1}\\end{cases}$$

\\(a_n = ?\quad^{\quad}\\)

Explanation:

Step1: Identify the sequence type

This is a geometric sequence since each term \(a_n\) is \(2\) times the previous term \(a_{n - 1}\). The first term (when \(n = 0\)) is \(a_0=5\).

Step2: Recall the formula for geometric sequence

The general formula for a geometric sequence with first term \(a_0\) and common ratio \(r\) is \(a_n=a_0\times r^n\). Here, \(a_0 = 5\) and the common ratio \(r = 2\) (since \(a_n=2a_{n - 1}\)).

Step3: Substitute the values

Substitute \(a_0 = 5\) and \(r = 2\) into the formula. So \(a_n=5\times2^n\).

Answer:

\(a_n = 5\times2^n\) (or in the boxed form as per the given structure, the first box is \(5\), the second box is \(2\), and the third box is \(n\))