Answer:
The series is represented by sum [tex]f(n) = 2 \cdot \Sigma\limits_{i=0}^{n} n[/tex], [tex]n \in\mathbb{N }[/tex]. The remaining element of the series is 30.
Step-by-step explanation:
This exercise consist in deriving the function which contains every element of the given series. The sum that contains all elements of the series { 0, 2, 6, 12, 20,...} is represented by the following formula:
[tex]f(n) = 2 \cdot \Sigma\limits_{i=0}^{n} n[/tex], [tex]n \in\mathbb{N }[/tex] (1)
Where [tex]n[/tex] is the cardinal associated with the element of the series.
Lastly, we proceed to evaluate the sum for the first five elements:
n = 0
[tex]f(0) = 0[/tex]
n = 1
[tex]f(1) = 0 +2[/tex]
[tex]f(1) = 2[/tex]
n = 2
[tex]f(2) = 0 + 2 + 4[/tex]
[tex]f(2) = 6[/tex]
n = 3
[tex]f(3) = 0 + 2 + 4 + 6[/tex]
[tex]f(3) = 12[/tex]
n = 4
[tex]f(4) = 0+2+4+6+8[/tex]
[tex]f(4) = 20[/tex]
The series is represented by sum [tex]f(n) = 2 \cdot \Sigma\limits_{i=0}^{n} n[/tex], [tex]n \in\mathbb{N }[/tex].
Lastly, the missing element is found by evaluating the function at [tex]n = 5[/tex], that is:
n = 5
[tex]f(5) = 0 + 2 + 4 + 6 + 8 + 10[/tex]
[tex]f(5) = 30[/tex]
The remaining element of the series is 30.
