Arithmetic Sequence

Arithmetic Sequence is any sequence of numbers in which consecutive numbers vary by a fixed difference. The first formal study of these sequences was said to be carried out by Carl Gauss, although his first contributions remain largely anecdotal. Gauss is said to have calculated the sum of the first 100 integers when he was 9 without any pen or paper. Although there is no evidence of this story, Gauss did make large use of arithmetic progression sequences in his study of discrete mathematics. Mathematicians such as Paul Turán, Erdős, Endre Szemerédi made further contributions by hypothesizing and proving several theorems related to arithmetic progressions/Sequence.

Definitions Arithmetic Sequence

Arithmetic progression or Arithmetic Sequence can be officially defined as an arithmetic sequence or progression is defined as a sequence of numbers in which for every pair of consecutive terms, the second number is obtained by adding a fixed number to the first one.
The common difference is defined as the fixed number that has to be added to any term in an arithmetic progression in order to obtain the immediately next number in the sequence.
First term of an arithmetic progression is the number with which the series starts. The first term is usually defined while defining the series. It is not necessary that an arithmetic progression has the first term.

The general form of an arithmetic progression is a1, a2, a3, ……………., an, where a2 – a1 is a common difference.

The last term of an arithmetic progression can be defined when the progression is being defined. It is not necessary for an arithmetic progression to have a last term. Infinite arithmetic series is an arithmetic progression where the first and last terms are not defined. The series extends from negative infinity to positive infinity.

Notation Arithmetic Sequence

A general arithmetic progression sequence is denoted by a1, a2, a3, ……………., an, where a2 – a1. The letter ‘a’ denotes a distinct arithmetic sequence. The subscript of the letter denotes the serial number of the term. For example,  a10, denotes the 10th term in the arithmetic sequence, and a1 denotes the first term of the sequence. The letter ‘d’ denotes the common difference of an arithmetic progression. Thus consecutive terms can be represented as  a2 = a1 + d and so on. Sn is used to denote the sum of an arithmetic progression from the starting term to the nth term. The value of n is variable. This notation is not universal. Several sources use different notations for the same terms. But the notations mentioned here are generally widely used.

Important formulae

Here are some formulae that are crucial in order to solve arithmetic progression related problems.

an = a + (n − 1) × d
This formula is used to determine the nth term of an arithmetic series.

S = n/2[2a + (n − 1) × d]
This formula is used to find the sum of an arithmetic series up to n terms.

S = n/2 (first term + last term)
This formula is used to find the sum of an arithmetic series when the value of the nth term is known.

Various other formulae are used to find other parameters. Those formulae can be derived from the above formulas or from the basic definitions of the arithmetic progressions.

Real life applications of Arithmetic Sequence

Various real life problems have been solved using arithmetic progressions. Arithmetic progressions and any series theory, in general, can be applied to any issue that shows a pattern.
People have predicted the eruptions of Old Faithful, an active geyser in Yellowstone national park using arithmetic progressions.
Taxi fares for certain distances are also calculated using arithmetic progressions. These methods are used by apps such as OLA and Uber to predict taxi fares.