Arithmetic Sequences 

t_n= t₁ + (n-1)d 

t_n= general term

t₁= first term

n= term number

d= constant difference 

Example 1: 5, 8, 11, 14…     

Since the sequence is going up by 3, the constant difference is 3.  d=3

 Find the 17^th term in the sequence.                                Find the general formula. 

t_n= t₁ + (n-1)d                                                              t_n= t₁ + (n-1)d

t₁₇= 5 + (17-1)3                                                           t_n= 5 + (n-1)3

t₁₇= 5 + (16)3                                                               t_n= 5+ 3n-3

t₁₇= 5 + 48                                                                   t_n= 2 + 3n

t₁₇= 51 

Example 2:  If the pattern is going down, the constant difference will be negative. 

-3, -10, -17, -24… 

Since the sequence is going down by 7, the constant difference is -7.  d= -7 

Find the 10^th term in the sequence.                                Find the general formula. 

t_n= t₁ + (n-1)d                                                              t_n= t₁ + (n-1)d

t₁₀= -3 + (10-1)(-7)                                                      t_n= -3 + (n-1)(-7)

t₁₀= -3 + (9)(-7)                                                           t_n= -3 + -7n +7

t₁₀= -3 + -63                                                                t_n= 4 + -7n

t₁₀= -66 

Geometric Sequences 

t_n= t₁ (r)^n-1 

t_n= general term

t₁= first term

r= common ratio

n= term number 

You find the common ratio for geometric sequences by dividing neighboring terms.   

Example 3:

5, 10, 20, 40… 

Find r:   10/5= 2        20/10=2        40/20=2       r=2

Notice that if the pattern is getting larger, that the absolute value of r is greater than 1. 

Example 4: 

64, -48, 36, -27… 

Find r:  -48/64= -3/4            36/-48= -3/4          -27/36= -3/4      r=-3/4

Notice that if the pattern switches positive, negative, positive, negative… or negative, positive…, that r is negative.  Notice that if the pattern is getting smaller that the absolute value of r is less than one. 

Example 5: 

200, -100, 50, -25… 

Find r:   -100/200= -1/2,   50/-100= -1/2,   -25/50= -1/2,   r= -1/2 

Find the 7^th term.                                                          Find the general formula

t_n= t₁ (r)^n-1                                                   t_n= t₁ (r)^n-1

t₇= 200 (-1/2)^7-1                                                            t_n= 200 (-1/2)^n-1

t₇= 200 (-1/2)⁶

t₇= 200 (1/64)

t₇= 50/16 or 3.125   

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