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Arithmetic Sequences

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 UNIT 10 : SEQUENCES AND SERIES

 LESSON 2: GEOMETRIC SEQUENCES HOMEWORK QUESTIONS

 

Quick Review

 

Text Box: a = t1 = first term
r = common ratio
n = term number
 

 

 

 

 

 

 

 


Homework Questions:

 

1. State which of the following are geometric. Find a and r for those that are geometric.

a) -1, 2, -4, 8,

c) -8, -4, 0, 4,

d) k, kx2, kx4,

 

2. In each of the following the general term is given. Determine the first 3 terms and find a, r, t4 and t7.

 

3. In each of the following geometric sequences, determine t7 and tn .

a) 4, 16, 64,

c) 100, 50, 25, 12.5,

e) 243, -81, 27,

 

 

5. Find the number of terms for the sequence below.

2, 6, 18, , 486

 

 

7. A car is purchased for $20 000.00. The value of the car depreciates 15% each year. Find its value in 7 years.

 

8. The population of a certain city is now 80 000. Each year it is projected to increase by 3%. Determine the population if 25 years.

 

9. A bacteria culture doubles every 5 min. If the initial count is 6 bacteria, how many will there be after 2 hours ?

 

 

11. Find the value of a that makes the following sequence geometric.

3, 12, 2a-4,

 

12. Find the value of x that makes the following sequence geometric.

x + 2, x + 4, 2x + 11,

 

 

Solutions:

 

1. State which of the following are geometric. Find a and r for those that are geometric.

a) -1, 2, -4, 8,

c) -8, -4, 0, 4,

d) k, kx2, kx4,

Solutions:

The sequence has a common ratio. Hence it is geometric with a = -1 and r = - 2.

 

The sequence has a common ratio. Hence it is geometric with a = 1 and r = - 1/3.

 

The sequence does not have a common ratio. Hence it is not geometric.

 

The sequence has a common ratio. Hence it is geometric with a = k and r = x2.

 

2. In each of the following the general term is given. Determine the first 3 terms and find a, r, t4 and t7.

Solutions:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3. In each of the following geometric sequences, determine t7 and tn .

a) 4, 16, 64,

c) 100, 50, 25, 12.5,

e) 243, -81, 27,

 

Solutions:

a = 12

r = 7

t7 = ?

tn = ?

 

a = 4

r =

t7 = ?

tn = ?

 

 

 

 

 

a = 100

r =

t7 = ?

tn = ?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5. Find the number of terms for the sequence below.

2, 6, 18, , 486

Solution:

a = 2

r = 3

n = ?

tn = 486

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


6. Given the geometric sequence 54, 36, 24, . Which term is 64/9 ?

Solution:

a = 5

r = -3

n = ?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


7. A car is purchased for $20 000.00. The value of the car depreciates 15% each year. Find its value in 7 years.

Solution:

 

a = 20000

r = 0.85

n = 8

tn = ?

 

 

8. The population of a certain city is now 80 000. Each year it is projected to increase by 3%. Determine the population if 25 years.

Solution:

The city retains 100% of its people and gains 3%. Therefore to get the next years population, multiply by 103% (1.03 in decimal form)

Hence the population in 25 years will be about 167 500.

 

9. A bacteria culture doubles every 5 min. If the initial count is 6 bacteria, how many will there be after 2 hours.

Solution:

 

 

10.  Radioactive decay is measured in terms of half-lives. This is the time it takes for a radioactive element to decay to half of its original mass.

Radioactive Geigerite has a half-life of 8 years. How long will it take for an original sample of 128 g to decay to 0.5 g?

 

Solution:

This forms a geometric sequence with first term 128 and common ratio .

 

 

11. Find the value of a that makes the following sequence geometric.

3, 12, 2a-4,

Solution:

Recall to be geometric, the test is

 

12. Find the value of x that makes the following sequence geometric.

x + 2, x + 4, 2x + 11,

Solution:

Recall to be geometric, the test is

 

Proof:

 

 

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