Transcript Arithmetic Sequence

Arithmetic Sequence
Chapter 2, lesson C
IB standard

Students should know Arithmetic sequence
and series; sum of finite arithmetic series;
geometric sequences and series; sum of
finite and infinite geometric series


Examples of applications, compound interest and
population growth
Sigma notation
Arithmetic Sequences


An arithmetic sequence is a sequence in
which each term differs from the pervious one
by the same fixed number
Example

2,5,8,11,14


5-2=8-5=11-8=14-11 etc
31,27,23,19

27-31=23-27=19-23 etc
Algebraic Definition

{Un} is arithmetic  Un+1 – Un= d for all
positive integers n where d is a constant (the
common difference)


 “If and only if”
{Un} is arithmetic then Un+1 – Un is a constant and
if Un+1 – Un is constant the {Un} is arithmetic
The General Formula



U1 is the 1st term of an arithmetic sequence
and the common difference is d
Then U2 = U1 + d therefore U3 = U1 + 2d
therefore U4 = U1 + 3d etc.
Then Un = U1 + (n-1)d
the coefficient of d is one less than the
subscript
Arithmetic Sequence

For arithmetic sequence with first term u1 and
common difference d the general term (or the
nth term) is un = u1 + (n-1)d
Examples #1

Consider the sequence 2,9,16,23,30…




Show that the sequence is arithmetic
Find the formula for the general term Un
Find the 100th term of the sequence
Is 828, 2341 a member of the sequence?
The middle term
If a, b, c are any consecutive terms of an
arithmetic sequence the
b - a= c - b (equating common differences)
therefore 2b= a+c
therefore b = (a+c) / 2
Thus the middle term is the arithmetic mean
(average) of terms on each side of it
- Hence the name arithmetic sequence

Example #2

Find k given that 3k+1 and -3 are consecutive
terms of an arithmetic sequence
Example #3


Find the general term Un for an arithmetic
sequence given that U3 = 8 and U8 = -17
Un = U1 + (n-1)d
Example #4

Insert four numbers between 3 and 12 so that
all six numbers are in arithmetic sequence.
Homework

Page 42-44 #1-9