PROOFS OF SOME FORMULAS USED FOR GEOMETRIC SEQUENCES AND SERIES
Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to firstname.lastname@example.org.
FINDING THE GENERAL TERM OF A GEOMETRIC SEQUENCE an= 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
an= 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:
Sn = a1 + a1r + a1r 2 + ... + a1r n - 2 + a1r n - 1
Now, multiply both sides of the equation by r:
rSn = a1r + a1r 2 + a1r 3 + ... + a1r n - 1 + a1r n
We will now subtract the second equation for the first equation to get
SnrSn = a1 a1r n
We are now going to factor out Sn on the left and a1 on the right to get
Sn(1 r) = a1 (1 r n)
Finally, we will solve for Sn 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 Sn 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.