The formula which represents the partial sum of the first n terms of the series is [tex]2n^{2}+\frac{1}{2}n[/tex] ⇒ D
Step-by-step explanation:
The sum of nth terms of the arithmetic series is [tex]S_{n}=\frac{n}{2}[a+l][/tex] , where
∵ The first n terms of the series are 2 + 6 + 10 + ......... + (4n - 1)
∵ 6 - 2 = 4 and 10 - 6 = 4
∴ There is a constant difference between the consecutive terms
∴ The series is arithmetic
∴ The sum of nth terms is [tex]S_{n}=\frac{n}{2}[a+l][/tex]
∵ The first term is 2
∴ a = 2
∵ The last term is (4n - 1)
∴ l = (4n - 1)
- Substitute these values in the rule
∴ [tex]S_{n}=\frac{n}{2}[2+(4n-1)][/tex]
∴ [tex]S_{n}=\frac{n}{2}[2+4n-1][/tex]
- Add like terms in the right hand side
∴ [tex]S_{n}=\frac{n}{2}[4n+1][/tex]
- Simplify it
∴ [tex]S_{n}=(\frac{n}{2})(4n)+(\frac{n}{2})(1)[/tex]
∴ [tex]S_{n}=2n^{2}+\frac{1}{2}n[/tex]
The formula which represents the partial sum of the first n terms of the series is [tex]2n^{2}+\frac{1}{2}n[/tex]
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You can learn more about the sequences in brainly.com/question/7221312
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