Answer:
[tex]S_1 = 12
\\ S_2 = 54
\\ S_3 = 126 \\ S_n = 15n^2 - 3n [/tex]
Explanation:
[tex]S_1 = 12
\\ S_2 = 12 +42 = 54
\\ S_3 = 12 + 42 + 72 = 126[/tex]
To calculate for [tex]S_n[/tex], note that the series is an arithmetic series because each term is equal to the preceding term plus a constant, which is 30 (called common difference).
We let
[tex]a_1 = \text{ first term} = 12 \\ a_2 = \text{ second term} = 42 \\ a_3 = \text{ third term} = 72\\.
\\.
\\.
\\a_n = n\text{th term}[/tex]
Since [tex]S_n[/tex] is an arithmetic series, [tex]a_1, a_2, a_3, ... , a_n[/tex] is an arithmetic sequence and so the formula for the nth term is given by
[tex]a_n = a_1 + d(n - 1)
\\ a_n = 12 + 30(n - 1)
\\ \boxed{a_n = 30n - 18}[/tex]
Where d = common difference = 30
Now, since [tex]S_n[/tex] is an arithmetic series, we can use the formula for the sum of the arithmetic series, which is given by
[tex]S_n = \frac{n}{2} (a_1 + a_n)
\\
\\ = \frac{n}{2} (12 + 30n - 18)
\\
\\= \frac{n}{2} (30n - 6)
\\
\\ = \frac{30n^2 - 6n}{2}
\\
\\ \boxed{S_n = 15n^2 - 3n}[/tex]
