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LammettHash

The underlying arithmetic progression is

-15, 0, 15, 30, ...

starting with [tex]a_1=-15[/tex] and with a common difference of [tex]d=15[/tex] between terms. Recursively, this sequence is given by

[tex]\begin{cases}a_1=-15\\a_n=a_{n-1}+15&\text{for }n>1\end{cases}[/tex]

So we have

[tex]a_2=a_1+15[/tex]

[tex]a_3=a_2+15=a_1+15\cdot2[/tex]

[tex]a_4=a_3+15=a_1+15\cdot3[/tex]

and so on, so that the explicit rule for the sequence is

[tex]a_n=a_1+15(n-1)=15n-30[/tex]

for [tex]n\ge1[/tex].

The series consists of 15 terms, since

[tex]195=15n-30\implies15n=225\implies n=15[/tex]

So we have

[tex]-15+0+15+\cdots+195=\displaystyle\sum_{n=1}^{15}(15n-30)=15\cdot\frac{15\cdot16}2-30\cdot15=1350[/tex]

and the answer would be B.

MrRoyal

The sum of the finite arithmetic series is 1350

The arithmetic series is given as:

(-15)+0+15+30+...+195

The above series have the following properties

First term (a) = -15

Common difference (d) = 15

Last term (L) = 195

The sum of a finite arithmetic series is:

[tex]S_n = \frac n2 * (a + L)[/tex]

Where:

[tex]L =a + (n - 1)d[/tex]

So, we have:

[tex]195 =-15 + (n - 1)15\\\\\\[/tex]

Add 15 to both sides

[tex]210 = (n - 1)15[/tex]

Divide both sides by 15

[tex]14 = n - 1[/tex]

Add 1 to both sides

n =15

So, we have:

[tex]S_n = \frac n2 * (a + L)[/tex]

[tex]S_{15} = \frac{15}{2} * (-15 + 195)[/tex]

[tex]S_{15} = 1350[/tex]

Hence, the sum of the finite arithmetic series is 1350

Read more about arithmetic series at:

https://brainly.com/question/7882626