Thomas O'Sullivan's Mathematics & Physics Pages
Sequences & Series Formulae
Arithmetic Progressions 
Geometric Progressions 
Example: 3, 7, 11, 15, ... The first term = a = 3 The common difference = d = 4 
Example: 3, 6, 12, 24, ... The first term = a = 3 The common ratio = r = 2 
d = u _{n}  u _{n1} 

A general Arithmetic Progression: a, a + d, a + 2d, a + 3d, .... , a + (n  1)d 
A General Geometric Progression: a, a r ^{2}, a r ^{3}, a r ^{4}, ... , a r ^{n1} 
u _{n }= a + (n  1)d 
u _{n }=_{ }a r ^{n1} 
To find the sum of an Arithmetic Progression (S _{n}): S_{n} = a + a + d + a +2d + ... + a+(n1)d S_{n} = a+(n1)d + a+(n2)d + a+(n3) + ... + a 2Sn = 2a+(n1)d +2a+(n1)d + 2a+(n1)d +... + 2a+(n1)d 2Sn = n(2a + n1)d 
To find the sum of a Geometric Progression (S _{n}) S _{n} = a + a r + a r^{2} + a r^{3} + ...+ a r ^{n1} r S _{n} = a r + a r^{2} + a r^{3} + ...+ a r ^{n1} + a r ^{n } S _{n}  r S _{n} = a  a r ^{n} S _{n} (1  r) = a(1  r ^{n}) 
3 Consecutive Terms of an arithmetic Progression x  d, x, x + d 
3 Consecutive Terms of a Geometric Progression 
To prove a sequence is arithmetic: Show that u _{n}  u _{n1 }= a constant (i.e. u _{n}  u _{n1 }is independent of n) 
To prove a sequence is geometric: Show that = a constant (i.e. independent of n) 
To find u _{n} given S _{n} U _{n} = S _{n}  S _{n1} 
To find u _{n} given S _{n} U _{n} = S _{n}  S _{n1} 
To find a and d when given two terms: e.g. If u _{2} = 5 and u _{4} = 11 find a and d. u _{2 }= a + (2  1)d u _{4} = a + (4  1)d 5 = a + d 11 = a + 3d Simultaneous equations gives a = 3 and d = 2 
To find a and r when given two terms: e.g. If u _{3} = 2 and u _{6} = 16 find a and r. u_{3} =a r ^{2}_{ }= 2 u_{6 }= a r ^{5}_{ }= 16 r = 2 and a = 1 
Arithmetic mean The arithmetic mean of two numbers is that number that when placed between two numbers, forms three consecutive terms of an arithmetic progression. e.g. the arithmetic mean of the numbers a and b is 
Geometric Mean If a, b and c are in geometric progression then 
Sum to infinity of a geometric progression: If r < 1 then 
© Thomas G. O'Sullivan 1999