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 SEQUENCEa_{n}= a_{1}r^{ n - 1}Consider a geometric sequence whose first term is

aand whose common ratio is_{1}r. Let's begin by writing the first six terms.

a_{1}a_{1}r a_{1}r^{ 2}a_{1}r^{ 3}a_{1}r^{ 4}a_{1}r^{ 5}

a_{1}a_{2}a_{3}a_{4}a_{5}a_{6}Compare each power of

rand the associated subscript ofa. The power ofris 1 less than the associated subscript ofa.Therefore, the formula for the

nthterm must be

a_{n}= a_{1}r^{ n - 1}which is the general term of a geometric sequence.

FINDING THE SUM OF N-TERMS OF A GEOMETRIC SEQUENCEStart with the sum of the first

nterms:

S=_{n}a_{1}+a_{1}r + a_{1}r^{ 2}+ ... + a_{1}r^{ n - 2}+ a_{1}r^{ n - 1}Now, multiply both sides of the equation by

r:

rS=_{n}a_{1}r +a_{1}r^{ 2}+ a_{1}r^{ 3}+ ... + a_{1}r^{ n - 1}+ a_{1}r^{ n}We will now subtract the second equation for the first equation to get

S_{n}rS=_{n}a_{1}a_{1}r^{ n}We are now going to factor out

Son the left and_{n}aon the right to get_{1}

S=_{n}(1 r)a_{1}(1r^{ n})Finally, we will solve for

Sto get_{n}which is the sum of the first

nterms of a geometric sequence.

FINDING THE SUM OF THE TERMS OF AN INFINITE GEOMETRIC SEQUENCEWe will first investigate what happens to

rin the formula if^{n}nis allowed to get larger and larger.We must understand that the term

rwill get smaller and smaller and will approaches 0, BUT only as long as^{n}ris any number between - 1 and 1.For example, observes what happens to

rwhen^{n}ngets larger and larger given .We see that

rgets smaller and smaller ultimately approaching 0.^{n}So let us replace

rwith 0 in the formula . We will get . Note that we no longer write^{n}Sbecause there is no_{n}nin the formula!is the formula for the sum of the terms of an infinite geometric sequence AS LONG AS

ris between 1 and 1.If

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