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Arithmetic Series

A sequence is referred as a set of numbers with a predefined order. These numbers are separated by commas.
For example:

  • 1, 4, 9, 16, 25, ....
  • 5, 10, 15, 20, 25, ....
Series
In mathematics, series is the sum of a finite sequence or we can say that it is a sum of few terms of a sequence.

Arithmetic Sequence
In arithmetic sequence or arithmetic progression, the difference between two consecutive numbers always remains same.
For example:
  • 1, 3, 5, 7, 9, .... where difference between 3 and 1, 5 and 3, 7and 5, 9 and 7 are all equal to 2.
  • 100, 97, 94, 91, .... where 97 -100 = 94 - 97 = 91 - 94 = - 3
Here, first term of the sequence is denoted by "a" and difference is denoted by "d" and is called "common difference".

Arithmetic Series
Arithmetic series is the sum of the finite terms of an arithmetic sequence, i.e. the sum of n number of terms of an arithmetic sequence is known as arithmetic series. When addition symbol (+) is used to separate every two consecutive numbers of an arithmetic sequence, it becomes arithmetic series.
For example, arithmetic series of above arithmetic sequences can be written as:
  • 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21
  • 100 + 97 + 94 + 91 + 88
We are able to find the value of sum of arithmetic series by the use of formula. The following formulas are used to calculate the sum of arithmetic series -

(1) Formula when first and last terms of the series are known:
$S_{n}=$$\frac{n}{2}$$(a+a_{n})$
Where,
n = Total number of terms
$S_{n}$ = Sum of first n terms
a = First term
$a_{n}$ = Last term

(2) Formula when first term and common difference are known:
$S_{n}=$$\frac{n}{2}$$[2a+(n-1)d]$
Where,
a = First term
d = Common difference
n = Total number of terms
$S_{n}$ = Sum of first n terms

Example: Estimate the sum of 10 terms of an arithmetic series 4 + 7 + 10 + ....
Solution: Here, a = 4, n = 10
and d = 7 - 4 = 10 - 7 = 3
Let us use the formula

$S_{n}=$$\frac{n}{2}$$[2a+(n-1)d]$

$S_{10}=$$\frac{10}{2}$$[2 \times 4 +(10-1) \times 3]$

$S_{10}=5 \times(8 + 27)$
Required sum = 175

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