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

Solve	Step	Spot Check
d  = 8 - 5 = 3,       d  = 11 - 8 = 3  d  = 14 - 11 = 3,   d  = 17 - 14 = 3	Find the common difference d = 3	a1 =  3(1) + 2 = 5
an = a1 + (n - 1)d	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
d  = 6 - 2 = 4,       d  = 10 - 6 = 4	Find the common difference d = 4	a1 =  4(1) - 2 =  2
an = a1 + (n - 1)d	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)n  Let a0 = initial value.
Solve	Step
a0 = 10,000	a0 = initial value
r = 0.12 ÷ 4 = 0.03	r  = 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