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UNIT 10  :  SEQUENCES AND SERIES

LESSON 2: GEOMETRIC SEQUENCES HOMEWORK QUESTIONS

Quick Review

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 = ¼ t7 = ? tn = ?

 a = 100 r = ½ 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 ½.

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: