Question

Which represent geometric sequences? A marathon runner runs half of a race, then runs half the distance left to the finish line, then runs half the distance that still remains, and so on. The first movie rental costs $2, and each additional rental costs $1. 10, 1, 0.1, 0.01, … –4, 20, –100, 500, … –1, 5, 11, 17, 23, …

Answer 1

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,

   $\frac{x}{2},\frac{x}{4},\frac{x}{8}.........$

  Common ratio = $\frac{\frac{x}{4}}{\frac{x}{2}}$

                           = $\frac{x}{4}\times \frac{2}{x}$

                           = $\frac{1}{2}$

  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 = $\frac{1}{10}=0.1$

  Ratio in 3rd and 2nd term = $\frac{0.1}{1}=0.1$    

  Therefore, ratio in each term is common.

  Sequence formed will be a geometric sequence.

4). $\sum_{k=1}^{7}4(0.2)^{k-1}$

   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 = $\frac{20}{-4}$ = -5

   Ratio of 3rd and 2nd term = $\frac{-100}{20}$

                                               = -5

   There is a common ratio = -5

   Therefore, it's a geometric sequence.

6). -1, 5, 11, 17.........

   Ratio of 2nd and 1st term = $\frac{5}{-1}=-5$

   Ratio of 3rd and 2nd term = $\frac{11}{5}=2.2$

   -5 ≠ 2.2

   Therefore, sequence is not a geometric sequence.

7). -500, 100, -20, 4........

   Ratio of 2nd and 1st term = $\frac{100}{-500}=-\frac{1}{5}$

   Ratio of 3rd and 2nd term = $\frac{-20}{100}=-\frac{1}{5}$