Geometric Sequences & Series

n = the number of terms in the sequence (n is a counter = 1,2,3,...)
r = the common ratio
t_n = the value of the nth term in the sequence
t₁ = the value of the first term in the sequence
S_n = the sum of the first n terms in the sequence

To find a term in the geometric sequence, use t_n = t₁r ^n-1.

To graph a sequence, n is on the x-axis and the value of the term is on the y-axis.

Example:

• Graph the sequence 2,4,8,16,..., find a formula for t_n, and find t₈.

       The graph is a set of points:  (1,2), (2,4), (3,8), (4,16),...
• t_n = t₁r^n-1

    = 2(2)^n-1   (the first term is 2; the common ratio is 2)
    = 2ⁿ
• t₈ = 2⁸ = 256   (applying the formula just found)

           S_n = t₁ (1 - rⁿ)
                      1 - r

A geometric series will converge if the common ration r is a proper fraction.  In other words, |r| < 1 or -1 < r < 1.

The sum of n terms of a series is given by S = t₁(1 - rⁿ)
                                                                         1 - r

If |r| < 1, then r_n ® 0 as n ® oo

and S ®    t₁     
                1 - r

So the series S will converge to   t₁
                                                1 - r