The sum of the series are:
Part(a):
Part(b):
Part(c):
Further explanation:
A series is defined as a sum of different numbers in which each term is obtained from a specific rule or pattern.
In this question we need to determine the sum of the series given in the part (a), part (b) and part (c).
Part(a):
The series given in part (a) is as follows:
All the terms in the given series are odd numbers.
From the given series in part(a) it is observed that the series is an arithmetic series with the common difference of .
An arithmetic series is a series in which each successive member of the series differs from its previous term by a constant quantity.
From the above series it is observed that the first term is , second term is , third term is , fourth term is , fifth term is and the last term is .
The nth term in a arithmetic series is given as follows:
(1)
In the above equation a represents the first term, represents the total terms and represents the common difference.
Substitute for , for and for in equation (1).
Therefore, total number of terms in the series is . This implies that .
The sum of an arithmetic series is calculated as follows:
(2)
Substitute for in equation (2).
Therefore, the sum of the series for part(a) is .
Part(b):
The series given in part (b) is as follows:
Express the given series as follows:
The series is as follows:
It is observed that the above series is exactly same as the series given in the part(a) and the sum of the series of part(a) as calculated above is .
Therefore, sum of the series is i.e., .
The series is as follows:
From the above series it is observed that the series is an arithmetic series as the difference between each consecutive member is and the last term is .
Substitute for , for and for in equation (1).
This implies that .
To calculate the sum of substitute for in equation (2).
Therefore, sum of the series is .
Substitute for and for in equation (3).
Therefore, the sum of the series for part(b) is .
Part(c):
The series given in part(c) is as follows:
From the above series it is observed that it is an arithmetic series with common difference as , first term as and the last term as .
Substitute for , for and for in equation (1).
Substitute for in equation (2).
Therefore, the sum of the series for part(c) is .
Learn more:
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Answer details:
Grade: High school
Subject: Mathematics
Chapter: Series
Keywords: Series, sequence, arithmetic sequence, arithmetic series, 1+3+5+7+9+….+99, 1-2+3-4+5-6+7-8+….-100, -100-99-98-….-2-1-0+1+2+…..+48+49+50, sum of series, first term, common difference.