Today our topic of discussion is Arithmetic Series.
Arithmetic Series
Arithmetic Series
If the difference between any two adjacent terms of a series is always equal, it is called arithmetic series.
Example 1. 1 + 3 + 5 + 7 + 9 + 11 is a series. The first term of the series is 1, the second term is 3, the third term is 5 etc.
Here, second term first term = 3-1-2,
third term-second term = 5-3=2, fourth term – third term = 7 – 5 = 2 ,
fifth term – fourth term = 9-7-5, sixth term – fifth term = 11-9=2 Hence, the series is an arithmetic series.
In this series, the difference between two terms is called common difference. The common difference of the mentioned series is 2. The numbers of terms of the series are fixed. That is why the series is finite series. It is to be noted that if the terms of the series are not fixed, the series is called infinite series, such as, 1 + 4 + 7 + 10 +*** is an infinite series.
In an arithmetic series, the first term and the common difference are generally denoted by a and d respectively. Then by definition, if the first term is a, the second term is a + d , the third term is a + 2d etc. Hence, the series will be a + (a + d) + (a + 2d) +***
Determining common term of an arithmetic series
Let the first term of an arithmetic series be a and the common difference be d. Then the terms of the series are:
First term a = a + (1 – 1) * d
Second term = a + d = a + (2 – 1) * d
Third term = a + 2 * (d = a + (3 – 1) * d)
Fourth term = a + 3 * (d = a + (4 – 1) * d)
……. …….. …….
……. ……… …….
∴ n th term = a+ (n-1)d
This nth term is called common term of arithmetic series. If the first term of an arithmetic series in a and common difference is d are known, all the terms of the series can be determined successively by putting n = 1, 2, 3, 4 ,*** in the nth term.
Let the first term of an arithmetic series be 3 and the common difference be 2. Then nth term of the series is = 3 + (n – 1) * 2 = 2n + 1
Work: If the first term of an arithmetic series is 5 and common difference is 7, determine the first six terms, 22nd term, rth term and (2p + 1) th term.
Example 2. Of the series 5 + 8 + 11 + 14 +*** which term is 383 ?
Solution: The first term of the series a = 5 common difference d = 8 – 5 = 11-8-14-11=3
∴ It is an arithmetic series.
Let, nth term of the series = 383
We know that, nth term = a+ (n-1)d
a + (n – 1) * d = 383
or, 5 + (n – 1) * 3 = 383
or, 5 + 3n – 3 = 383
or, 3n = 383 – 5 + 3
or, 3n = 381
or, n = 381/3
or, n = 127
∴127 th term of the given series is 383
Sum of n terms of an arithmetic series
Let the first term of any arithmetic series be a, last term be p, common difference be d, number of terms be n and sum of n terms be Sn
Writing from the first term to the last and conversely from the last term to the first of the series we get,
Sn =a+(a+d)+(a+2d)+***+ (p – 2d) + (p – d) +p…(1)
and Sn = p + (p – d) + (p – 2d) +***+(a+2d)+(a+d)+a…(2)
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Adding, Sn = (a + p) + (a + p) + (a + p) +***+(a+p)+(a+p)+(a+p)
or, 2S_{n} = n(a + p) [ number of terms of the series is n]
∴ S n = n/2 * (a + p) …(3)
Again, nth term = p = a + (n – 1) * d . Putting this value of p in (3) we get,
S_{n} = n/2 * [a + \{a + (n – 1) * d\}]
i.e., S n = n/2 * {2a + (n – 1) * d} …(4)
If the first term of an arithmetic series a, last term p and number of terms n are known, the sum of the series can be determined by the formula (3). But if the first term a, common difference d and number of terms n are known, the sum of the series are determined by the formula (4).
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