Answer:
Step-by-step explanation:
1). Let the length of a marathon race = x miles
Marathon runner runs this race in the sequence formed as,
[tex]\frac{x}{2},\frac{x}{4},\frac{x}{8}.........[/tex]
Common ratio = [tex]\frac{\frac{x}{4}}{\frac{x}{2}}[/tex]
= [tex]\frac{x}{4}\times \frac{2}{x}[/tex]
= [tex]\frac{1}{2}[/tex]
Therefore, it's a geometric sequence.
2). First movie rental cost = $2
Followed by each additional movie rental cost = $1
Therefore, rental of movies will be in the form of a linear equation,
y = 2x + 1
where x = number of movies
And y = Rental of movies
It's not a geometric sequence.
3). 10, 1, 0.1, 0.01.......
Ratio in 2nd and 1st term = [tex]\frac{1}{10}=0.1[/tex]
Ratio in 3rd and 2nd term = [tex]\frac{0.1}{1}=0.1[/tex]
Therefore, ratio in each term is common.
Sequence formed will be a geometric sequence.
4). [tex]\sum_{k=1}^{7}4(0.2)^{k-1}[/tex]
Expression represents a common ratio of 0.2,
Therefore, expression given will represent a geometric sequence.
5). -4, 20, -100, 500, ...........
Ratio of 2nd and 1st term = [tex]\frac{20}{-4}[/tex] = -5
Ratio of 3rd and 2nd term = [tex]\frac{-100}{20}[/tex]
= -5
There is a common ratio = -5
Therefore, it's a geometric sequence.
6). -1, 5, 11, 17.........
Ratio of 2nd and 1st term = [tex]\frac{5}{-1}=-5[/tex]
Ratio of 3rd and 2nd term = [tex]\frac{11}{5}=2.2[/tex]
-5 ≠ 2.2
Therefore, sequence is not a geometric sequence.
7). -500, 100, -20, 4........
Ratio of 2nd and 1st term = [tex]\frac{100}{-500}=-\frac{1}{5}[/tex]
Ratio of 3rd and 2nd term = [tex]\frac{-20}{100}=-\frac{1}{5}[/tex]
