PROOFS OF SOME FORMULAS USED FOR GEOMETRIC SEQUENCES AND SERIES

Copyright by Ingrid Stewart, Ph.D.  Please Send Questions and Comments to ingrid.stewart@csn.edu.

FINDING THE GENERAL TERM OF A GEOMETRIC SEQUENCE a_n= a₁r ^{n - 1}

Consider a geometric sequence whose first term is a₁ and whose common ratio is r. Let's begin by writing the first six terms.

a₁                    a₁r        a₁r ²      a₁r ³      a₁r ⁴      a₁r ⁵

a₁                     a₂           a₃         a₄         a₅         a₆

Compare each power of r and the associated subscript of a. The power of r is 1 less than the associated subscript of a.

Therefore, the formula for the nth term must be

a_n= a₁r ^{n - 1}

which is the general term of a geometric sequence.

FINDING THE SUM OF N-TERMS OF A GEOMETRIC SEQUENCE

Start with the sum of the first n terms:

               S_n = a₁  + a₁r  + a₁r ² + ... + a₁r ^{n - 2}  +  a₁r ^{n - 1}

Now, multiply both sides of the equation by r:

               rS_n = a₁r  + a₁r ²  + a₁r ³ + ... + a₁r ^{n - 1}  +  a₁r ⁿ

We will now subtract the second equation for the first equation to get

                   S_nrS_n = a₁  a₁r ⁿ

We are now going to factor out S_n on the left and a₁ on the right to get

                 S_n(1 r) = a₁ (1 r ⁿ)

Finally, we will solve for S_n to get

which is the sum of the first n terms of a geometric sequence.

FINDING THE SUM OF THE TERMS OF AN INFINITE GEOMETRIC SEQUENCE

We will first investigate what happens to r ⁿ in the formula if n is allowed to get larger and larger.

We must understand that the term r ⁿ will get smaller and smaller and will approaches 0, BUT only as long as r is any number between - 1 and 1.

For example, observes what happens to r ⁿ when n gets larger and larger given .

We see that r ⁿ gets smaller and smaller ultimately approaching 0.

So let us replace r ⁿ with 0 in the formula . We will get . Note that we no longer write S_n because there is no n in the formula!

is the formula for the sum of the terms of an infinite geometric sequence AS LONG AS r is between 1 and 1.

If r is greater than or equal to 1, there is no sum of the infinite geometric squence.