4. 5, ____ , 45, -135, ...\ncommon ratio: r = ____\nmissing number: ____\nexplicit formula:____\nrecursive…

Explanation:

Step1: Find common ratio r

$r = \frac{-135}{45} = -3$

Step2: Calculate missing number

$5 \times (-3) = -15$

Step3: Derive explicit formula

For a geometric sequence, the explicit formula is $a_n = a_1 r^{n-1}$, where $a_1=5$, $r=-3$. $a_n = 5 \times (-3)^{n-1}$

Step4: Derive recursive formula

A geometric sequence recursive formula is $a_n = r \times a_{n-1}$ with $a_1$ defined. $a_1 = 5$, $a_n = -3 \times a_{n-1}$ for $n \geq 2$

Answer:

common ratio: $r = -3$ missing number: $-15$ explicit formula: $a_n = 5 \times (-3)^{n-1}$ recursive formula: $a_1 = 5$, $a_n = -3a_{n-1}$ for $n \geq 2$