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 n-1

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 n-1

u n = a + (n - 1)d

 

u n = a r n-1

To find the sum of an Arithmetic Progression (S n):

Sn = a + a + d + a +2d + ... + a+(n-1)d

Sn = a+(n-1)d + a+(n-2)d + a+(n-3) + ... + a

2Sn = 2a+(n-1)d +2a+(n-1)d + 2a+(n-1)d +... + 2a+(n-1)d

2Sn = n(2a + n-1)d

To find the sum of a Geometric Progression (S n)

S n = a + a r + a r2 + a r3 + ...+ a r n-1

r S n = a r + a r2 + a r3 + ...+ a r n-1 + 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 n-1 = a constant

(i.e. u n - u n-1 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 n-1

To find u n given S n

U n = S n - S n-1

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.

u3 =a r 2 = 2 u6 = 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



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© Thomas G. O'Sullivan 1999