PROOFS OF SOME FORMULAS USED FOR ARITHMETIC SEQUENCES AND SERIES

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

FINDING THE GENERAL TERM OF AN ARITHMETIC SEQUENCEa_{n}= a_{n}(n 1)dConsider an arithmetic sequence whose first term is

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

a_{1}a_{1}+ d a_{1}+ 2d a_{1}+ 3d a_{1}+ 4da_{1}+ 5d

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

dand the associated subscript ofa. The coefficient ofdis 1 less than the associated subscript ofa.Therefore, the formula for the

nthterm must be

a_{n}= a_{n}(n 1)dwhich is the general term of an arithmetic sequence.

FINDING THE SUM OF N-TERMS OF AN ARITHMETIC SEQUENCEWrite the sum of the terms starting with the first term

aand keep on adding_{1}das follows:

S=_{n}a+_{1}(a_{1}+ d) + (a_{1}+ 2d) + ... + a_{n}Write the sum of the terms, however, now starting with the last term

aand keep on subtracting_{n}das follows:

S=_{n}a+_{n}(a_{n}d) + (a_{n}2d) + ... + a_{1}We will now add the two equations to get

2S=_{n}a+_{1}a+_{n}(a_{1}+a) +_{n}(a+ ... +_{1}+a)_{n}(a_{1}+a)_{n}Because there are

nsums of(aon the right, we can express this side as_{1}+a)_{n}n(a. Therefore, the equation can be written as_{1}+a)_{n}

2S=_{n}n(a_{1}+a)_{n}and dividing both sides by 2, we get

which is the sum of the first

nterms of an arithmetic sequence.