Arithmetic, geometric, and harmonic progressions

Elements
Arithmetic Progression, AP
1. Formulas for Arithmetic Progression
Geometric Progression, GP
1. Formulas for Geometric Progression
2. Sum of Infinite Geometric Progression
Harmonic Progression, HP
Relationship between arithmetic, geometric, and harmonic means
Derivation of Formulas

Elements

a₁ = value of the first term
a_m = value of any term after the first term but before the last term
a_n = value of the last term
n = total number of terms
m = m^th term after the first but before n^th
d = common difference of arithmetic progression
r = common ratio of geometric progression
S = sum of the 1^st n terms

Arithmetic Progression, AP

Arithmetic progression is a sequence of numbers in which the difference of any two adjacent terms is constant. The constant difference is commonly known as common difference and is denoted by d. Examples of arithmetic progression are as follows:

Example 1: 3, 8, 13, 18, 23, 28 33, 38, 43, 48
The above sequence of numbers is composed of n = 10 terms (or elements). The first term a₁ = 3, and the last term a_n = a₁₀ = 48. The common difference of the above AP is d = 8 - 3 = 13 - 8 = ... = 5.

Example 2: 5, 2, -1, ...
This AP has a common difference of -3 and is composed of infinite number of terms as indicated by the three ellipses at the end.

Formulas for Arithmetic Progression

Common difference, d
The common difference can be found by subtracting any two adjacent terms.

$d = a_{m + 1} - a_m$ or

$d = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = ...$

Value of each term
Each term after the first can be found by adding recursively the common difference d to the preceding term.

$a_{m + 1} = a_m + d$

n^th term of AP
The n^th term of arithmetic progression is given by

$a_n = a_1 + (n - 1)d$

or in more general term, it can be written as

$a_n = a_m + (n - m)d$

Sum of n terms of AP
The sum of the first n terms of arithmetic progression is n times the average of the first term and the last term.

$S = \dfrac{n}{2}(a_1 + a_n)$

If the last term an is not given, the following may be useful

$S = \dfrac{n}{2}[ \, 2a_1 + (n - 1)d \, ]$

If required for the partial sum from m^th to n^th terms, the following formula can be used

$S = \dfrac{n - m + 1}{2}(a_m + a_n)$ or $S = \dfrac{n - m + 1}{2} [ \, 2a_m + (n - m)d \, ]$

Geometric Progression, GP

Geometric progression is a sequence of numbers in which any two adjacent terms has a common ratio denoted by r. Example of geometric progression is

1, 3, 9, 27, ...

which is composed of infinite number of terms and with common ratio equal to 3.

Formulas for Geometric Progression

Common ratio
The common ratio can be found by taking the quotient of any two adjacent terms.

$r = \dfrac{a_{m + 1}}{a_m} = \dfrac{a_2}{a_1} = \dfrac{a_3}{a_2} = \dfrac{a_4}{a_3} = ...$

n^th term of GP
The n^th term of the geometric progression is given by

$a_n = a_1 \, r^{n - 1}$ or $a_n = a_m \, r^{n - m}$

Sum of n terms of GP
The sum of the first n terms of geometric progression is

$S = \dfrac{a_1(1 - r^n)}{1 - r}$

Sum of Infinite Geometric Progression

A finite sum can be obtained from GP with infinite terms if and only if -1.0 ≤ r ≤ 1.0 and r ≠ 0.

$S = \dfrac{a_1}{1 - r}$

Harmonic Progression, HP

Harmonic progression is a sequence of numbers in which the reciprocals of the elements are in arithmetic progression. Example of harmonic progression is

1/3, 1/6, 1/9, ...

If you take the reciprocal of each term from the above HP, the sequence will become

3, 6, 9, ...

which is an AP with a common difference of 3.

Another example of HP is 6, 3, 2. The reciprocals of each term are 1/6, 1/3, 1/2 which is an AP with a common difference of 1/6.

To find the term of HP, convert the sequence into AP then do the calculations using the AP formulas. Then take the reciprocal of the answer in AP to get the correct term in HP.

Relationship between arithmetic, geometric, and harmonic means

$AM \times HM = GM^2$

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