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 SEQUENCE a_n= a_n(n 1)d

Consider an arithmetic sequence whose first term is a₁ and whose common difference is d. Let's begin by writing the first six terms.

a₁                    a₁+ d        a₁+ 2d      a₁+ 3d      a₁+ 4d      a₁+ 5d

a₁                       a₂               a₃             a₄             a₅             a₆

Compare each coefficient of d and the associated subscript of a. The coefficient of d is 1 less than the associated subscript of a.

Therefore, the formula for the nth term must be

a_n= a_n(n 1)d

which is the general term of an arithmetic sequence.

FINDING THE SUM OF N-TERMS OF AN ARITHMETIC SEQUENCE

Write the sum of the terms starting with the first term a₁ and keep on adding d as follows:

               S_n = a₁                   + (a₁+ d)       + (a₁+ 2d)     + ... + a_n

Write the sum of the terms, however, now starting with the last term a_n and keep on subtracting d as follows:

               S_n = a_n                   + (a_nd)       + (a_n2d)     + ... + a₁

We will now add the two equations to get

                   2S_n = a₁ + a_n           + (a₁+ a_n )    + (a₁+ a_n )    + ... + (a₁+ a_n )

Because there are n sums of (a₁+ a_n ) on the right, we can express this side as n(a₁+ a_n ). Therefore, the equation can be written as

             2S_n = n(a₁+ a_n )

and dividing both sides by 2, we get

which is the sum of the first n terms of an arithmetic sequence.