Arithmetic Sequences and Series

Rahul Chacko
IB Mathematics HL Revision – Step One
Chapter 1.1 – Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series. Sigma notation.
Arithmetic Sequences
Definition: An arithmetic sequence is a sequence in which each term differs from the previous one by the same fixed number:
{un} is arithmetic if and only if u n 1  u n  d .
Information Booklet u n  u1  n  1d

Proof/Derivation: u n 1  u n  d
 u n  u n 1  d
 u n 1  u1  dn
 u n  u1  dn
 u n  u1  n  1d
Derivations:
u1  u n  n  1d u  u1 d n n 1 u  u1 n n
1
d
Information Booklet

Sn 

n
2u1  n  1d   n u1  u n 
2
2

Proof:
Sn = u1 + u2 + u3 + …+ un
= u1 + (u1 + d) + (u1 + 2d) + (u1 + 3d) + …+ (u1 + (n − 1)d)
= un + (un − d) + (un − 2d) + (un + 3d) + …+ (un − (n − 1)d)
2Sn = n(u1 + un) n  S n  u1  u n 
2

Derivations

2S n
 u1 n 2S u1  n  u n n 2S n n u1  u n

un 

Geometric Sequences
Definition: A geometric sequence is a sequence in which each term can be obtained from the previous one by multiplying by the same non-zero constant.
{un} is geometric if and only if

u n 1
 r , n    where r is a constant. un Information Booklet u n  u1 r n 1

Proof: u n 1
r
un
 u n r  u n 1
 un 

 u n 1

u n 1 r  u1 r n

 u n  u1 r n 1
Derivations:
u u1  nn 1 r 
1

 u  n 1 r  n 
u 
 1 u n  log r n  1 u1 u log n u1 n
1
log r

(non-calculator paper)

(calculator paper)

Compound Interest:
100%  i %
, i  interest rate per
100%
compounding period, n = number of periods and u n 1  amount after n periods.

u n 1  u1 r n , where u1  initial investment,

r

Information Booklet

Sn 

u1 r n  1 u1 1  r n 
,r1
 r 1
1 r

Proof:

Sn = u1 + u2 + u3 + …+ un-1 + un
= u1 + u1r + u1r2 + u1r3 + … + u1rn−2 + u1rn−1
rSn =