1. 1.8 Geometric sequences and series
Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton 2012

2. Geometric sequences and series
A geometric sequence is one in which there is a common ratio (r)
between successive terms.
The sequences below are therefore geometric. ×2
r = 2
r =
Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton 2012

3. Geometric sequences and series
In general u1 = the first term of a geometric sequence
r = the common ratio
n = the number of terms
A geometric sequence can therefore be written in its general form as:
u1 u1r u1r2 u1r3 u1r4 … u1rn–2 u1rn–1
Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton 2012

4. Geometric sequences and series
A geometric series is one in which the sum of the terms of a
geometric sequence is found.
e.g. The sequence
can be written as a series as
The sum of this series is therefore 189.
The general form of a geometric series can therefore be written as:
u1 + u1r + u1r2 + u1r3 + u1r4 + … + u1rn–2 + u1rn–1
Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton 2012

5. Geometric sequences and series
The formula for the sum (Sn) of a geometric series can be deduced
as follows.
Sn = u1 + u1r + u1r2 + u1r3 + u1r4 + … + u1rn–2 + u1rn–1
Multiplying each term in the series by r gives
rSn = u1r + u1r2 + u1r3 + u1r4 + u1r5 + … + u1rn–1 + u1rn
Subtracting the second formula from the first gives
rSn = u1r + u1r2 + u1r3 + u1r4 + u1r5 + … + u1rn–1 + u1rn
rSn – Sn = –u u1rn
Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton 2012

6. Geometric sequences and series
The subtraction of one formula from the other was seen to give
rSn – Sn = –u1 + u1rn
Which in turn can be rearranged to give
rSn – Sn = u1rn – u1
Factorising gives
Sn(r – 1) = u1(rn – 1)
So the formula for the sum of n terms of a geometric series is
Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton 2012