Geometric Sequences

A geometric sequence, or geometric progression, is a sequence defined by u1 = a and u i+1 = rui, where r is not equal to 0 or 1.
Examples :

   {2, 4, 8, 16, 32} is a geometric term with six terms.
   {2, 1, 0.5, 0.25, 0.125, 0.0625, . . .} is an infinite geometric sequence.


I will skip the proving parts and jump directly to the formula gained from deriving the equations.

The sum of the geometric series a + ar + ar^2 + ... + ar^n-1, with n terms, is

S∞ is called the sum of infinity of the series.


This following examples are the typical exam-typed question. You should try it out.


Ex. Express the recurring decimal 0.296296296 ... as a fraction.

            The decimal can be written as
                        0.296 + 0.296 x 0.001 + 0.296 x (0.001)^2 + ... and so on
            which is a geometric series with a = 0.296 and r = 0.001. Since | r | < 1, the series iconvergent          with limiting sum 

Since 296 = 8 x 37 and 999 = 27 x 37, this fraction is in its simplest form 8/27.





u0 = a     and ui+1 = rui,
or by the formula

    ui = ar^i .

Example :
Saria's grandparents put $1000 into a savings bank account for her on each birthday from her 10th to her 18th. The account pays interest at 6% for each complete year that the money is invested. How much money is in the account on the day after her 18th birthday ?

Start with the most recent deposit. The $1000 on her 18th birthday has not earned any interest. The $1000 on her 17th birthday has earned interest for one year, so is now worth $1000 x 1.06 = $1060. Similarly, the $1000 on her 16th birthday is worth 41000 x 1.06^2 = $1124, and so on. So the total amount is now $S, where
           S = 1000 + 1000 x 1.06 + 1000 x (1.06)^2 + ... + 1000x(1.06)^8.

The sum is a geometric series with a = 1000 and r = 1.06 and n = 9. Using the general formula in the alternative version for r > 1,

I will upload more exercises with solutions soon.

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