Arithmetic and Geometric Sequences

-- Sections 8.3/8.4 --

 
Arithmetic Sequence: is a sequence with a common difference,  an = a1 + (n - 1)
where d is the common difference.
Geometric Sequence: is a sequence with a common ratio,  an = a1rn - 1 
where r is the common ratio.

  Find the nth term of the following arithmetic sequence   5, 8, 11, 14, 17, ...
  Then find the 100th term.

Solve

Step

          Spot Check

= 8 - 5 = 3,       = 11 - 8 = 3 
= 14 - 11 = 3,    = 17 - 14 = 3

Find the common difference
d = 3

a1 3(1) + 2 = 5

an = a1 + (n - 1)

Use the formula for arithmetic sequence since there is a common difference a3 3(3) + 2 = 9 + 2 = 11

a1 = 5 and   d = 3

a1 = 5 is the first term of the sequence. a5 3(5) + 2 = 15 + 2 = 17

an = 5 + (n - 1)3 

Plug in the values... 

an = 5 + 3n - 3 = 3n + 2

... and simplify.

an 3n + 2

nth term

a100 3(100) + 2 = 300 + 2 = 302

100th term  

  
  Find the nth term of the following arithmetic sequence   29, 25, 21, 17, 13, ...
  Then find the 25th term.

[Solution]

 Find the sum of the first 20 terms of the arithmetic sequence 2, 6, 10,... .

First find the nth term.  Then use the formula for the sum of an arithmetic sequence.
Solve Step

           Spot Check

= 6 - 2 = 4,       = 10 - 6 = 4 

Find the common difference
d = 4

a1 4(1) - 2 =  2

an = a1 + (n - 1)

Use the formula for arithmetic sequence.  a3 4(3) - 2 =  12 - 2 = 10

a1 = 2 and   d = 4

a1 = 2 is the first term of the sequence.

an = 2 + (n - 1)4 

Plug in the values... 

an = 2 + 4n - 4 = 4n - 2

... and simplify.

an 4n - 2

nth term
Use the formula for the sum of an arithmetic sequence. Alternate formula

n = number of terms
a = first term
l = last term
 

a = 2,  l = a20 4(20) - 2 = 80 - 2 = 78

n = 20

Plug in the values... and simplify.

 

 Find the sum of the first 30 terms of the arithmetic sequence 1, 5, 9,... .

[Solution]

   Find the nth term of the following geometric sequence   81, 27, 9, 3, 1, ...
  Then find the 7th term.
Solve Step

           Spot Check

Find the common ratio

r = 1/3

an = a1rn - 1 

Use the formula for geometric sequence. 

a1 = 81 and   r = 1/3

a1 = 81 is the first term of the sequence.

nth term

7th term

  Find the nth term of the following geometric sequence  1, 5, 25, 125, 625, ...
  Then find the 7th term.

[Solution]

Find the value of $10,000 left on deposit for 20 years at an annual rate of 12% compounded quarterly.

This is a geometric sequence problem with a small change.  an = a0(1+ r)Let a0 = initial value.

Solve Step

a0 = 10,000

a0 = initial value

r = 0.12 4 = 0.03 = 12% annual rate divided by 4 (quarterly is 4 times a year)
n = 20 4 = 80 n = number of years times by 4 (quarterly is 4 times a year)

a80 = 10,000(1 + 0.03)80 =

a80 = 10,000(1.03)80 » $106,408.91

The account would have $106,408.91 at the end of 20 years.

  Find the value of $20,000 left on deposit for 30 years at an annual rate of 9% compounded monthly.

[Solution]

Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada