Arithmetic Sequences and Series

I.  Homework

II.  Arithmetic Sequence

Exercise:  Find the general term for the following sequences both recursively and explicitly:

A)  2,6,10,14,18,22, ...

B)  -5,-3,-1,1,3,...

C)  1,4,7,10,13,16,...

D)  -1,10,21,32,43,54,...

E)  3,0,-3,-6,-9,-12,...

Definition:  A sequence with general term

a_n+1 = a_n + d

is called an arithmetic sequence.

Theorem: An arithmetic sequence with a_n+1 = a_n + d  has explicit form

a_n = a₁ + (n - 1)d

Proof:  (by induction)

For n = 1, we have a₁ = a₁ + (1 - 1)d (true)

Assume that the theorem is true for n = k - 1, hence

a_k-1 = a₁ + (k - 1 - 1)d = a₁ + (k - 2)d

Then a_k = a_k-1 + d = a₁ + (k - 2)d + d

= a₁ + kd - 2d + d = a₁ + kd - d = a₁ + (k - 1)d

Hence by MI the theorem is true.

Example:  Suppose that a₁ = 4 and d = 2 then the sequence is 4,6,8,...,(4 + (n - 1)d),...

Exercise:  Suppose that the 13th term of an arithmetic sequence is 46 and the fourth term is 100.  Find the expression for the general term.

III.  The Arithmetic Series

Last time, we proved that if

{a₁ + (n - 1)d}

is an arithmetic sequence then the sum of the sequence is

n/2 (a₁ + a_n)

Exercise:  

Find 3 + 7 + 11 + 15  + ... + 35

Exercise:  Suppose that the sum of the first 18 terms of an arithmetic sequence is -45 and d = -9, find the first term.

Application  Suppose that you play black jack at Harrah's on June 1 and lose $1,000.  Tomorrow you bet and lose $15 less.  What will your total losses be for the 30 days of June?