Find common difference or common ratio in a sequence. 
1) What is the common difference in the sequence
2m + 1, 4m + 4, 6m + 7, 8m + 10, ... ?
A "common difference" tells you that this is an arithmetic sequence where something is being added to each term to get the next term. To find what is being added, subtract any term from the term to its immediate right.
(4m + 4)  (2m + 1) = 2m + 3 is the common difference
2) What is the common ratio of the sequence
A "common ratio" tells you that this is a geometric sequence where something is being multiplied to each term to get the next term. To find what is being multiplied, divided any term by the term immediately in front (to the left) of it.

First,
Now,
remember the rules for working with exponents involving division. Subtract the exponents, place result in numerator, or place where larger exponent resides.
The common ratio is .

3) What is the common ratio of a geometric sequence whose first term is 27 and whose fourth term is 64?
If the term numbers are small enough, make a visual of the sequence.
From the visual we can see that 27 • r • r • r = 64, or 27r^{3} = 64.
(27 gets multiplied by the common ratio 3 times to arrive at 64)
The common ratio is 4/3.


Find a specific term in a sequence. 
1) Find the 12^{th} term of the sequence a_{n} = n(n + 2).
Substitute the number of the term into the sequence formula.
a_{12} = 12 (12 + 2) = 12 (14) = 168
2) Find the 15^{th} term of the sequence f (n) = (1)^{n1}n^{2}.
f (15) = (1)^{151}15^{2} = (1)^{14}15^{2} = 1•225 = 225
3) If f (1) = 3 and f (n) = 2f (n  1) + 1, then f (5) = _______.
This is a recursive formula where the formula is based upon the term that comes directly before the term you seek. To find the fifth term, we need to know the fourth term. To find the fourth term, you need to know the third term, and so on.
f (1) = 3
f (2) = 2(3) + 1 = 5
f (3) = 2 (5) + 1= 11
f (4) = 2(11) + 1 = 21
f (5) = 2(21) + 1 = 43
Analyze the problem and solve using sequence skills. 
1) The diagrams below represent the first three terms of a sequence, where the terms correspond to the number of shaded squares in each diagram.
Assuming that this pattern continues, which formula determines a_{n},
the number of shaded squares in the n^{th} term?
1) a_{n} = 6n + 4 2) a_{n} = 6n + 8 3) a_{n} = 6n + 10 4) a_{n} = 6n + 12
Make a list of the shaded squares so you can see the developing pattern. 
First term has 16 shaded squares.
Second term has 22 shaded squares.
Third term has
28 shaded squares. 
16, 22, 28, ... represents the sequence.
The sequence is arithmetic
with a common difference of 6.

Examine the sequence as it relates to n: 
When n = 1, the term is 16 = 6•1 + 10.
When n = 2, the term is 22 = 6•2 + 10.
When n = 3, the term is 28 = 6•3 + 10.
Only choice 3 gives this pattern.

2) A seedling is 3 inches tall and is predicted to grow an additional 4 inches each week. Which of the following formulas can be used to predict the height of the seedling at the end of n weeks, assuming this growth pattern continues?
I. a_{n} = 4n + 3
II.
a_{n} = 4n + 3(n  1)
III.
a_{1} = 7; a_{n} = a_{n1} + 4 
1) I and II 2) I and III 3) II only 4) III only
At the end of the first week, the seedling will be 7 inches tall. At the end of the second week it will be 11 inches tall and so on. Using the sequence to represent the height at the end of each week, we have 7, 11, 15, 19, ...
Glance at the choices. Choice I and choice II are not equivalent expressions, so they cannot both be true. By substitution of n = 1, 2, 3, 4, ..., choice I gives 7, 11, 15, 19, ... while choice II gives 4, 11, 18, 25, ...
Choice III is a recursive formula stating that the first term is 7 and each successive term is 4 more than the term in front of it. 7, 11, 15, 19, ...
Choices I and III represent the growth of the seedling.
3) Given a sequence whose n^{th} term is given by the explicit formula: a_{n} = 2n = 1
a) Write the first three terms of the sequence.
b) Find the 25^{th} term.
c) Find the number of the term whose value is 407.
d) Will there be a term whose value is 90? Explain.
SOLUTION:
a) Since n is a natural number (1, 2, 3, 4, ...), the first term of a sequence will begin with n = 1. 

a_{1} = 2(1)  1 = 1
a_{2} = 2(2)  1 = 3
a_{3} = 2(3)  1 = 5 
Notice that n is replaced with the number of the term
you are trying to find. 

b) a_{25} = 2(25)  1 = 49 
c) 2n  1 = 307
2n = 408
n = 204 The 204^{th} term. 
d) No.
2n  1 = 90
2n = 91
n = 45.5 (not a natural (counting) number)
You cannot have 45½ terms. The number of terms must be a counting number (1, 2, 3, ...).
