The sum of n terms of AP is the sum(addition) of the first n terms of the arithmetic sequence.
It is equal to n divided by 2 times the sum of twice the first term – ‘a’ and the product of the difference between second and first term - ‘d’ also known as common difference, and (n-1), where n is the number of terms to be added.
The formula is given to be:
[tex]S=\frac{n}{2}(2a+\lbrack n-1\rbrack d)[/tex]
From the series given, we have the following parameters:
[tex]\begin{gathered} a=12 \\ d=4-12=-8 \\ n=14 \end{gathered}[/tex]
Substituting these values into the formula, we have:
[tex]\begin{gathered} S_{14}=\frac{14}{2}(2\times12+\lbrack14-1\rbrack\times-8) \\ S_{14}=7(24+\lbrack13\times-8\rbrack) \\ S_{14}=7(24-104) \\ S_{14}=7\times-80 \\ S_{14}=-560 \end{gathered}[/tex]
The sum of the first 14 terms is -560.
The correct option is the THIRD OPTION.
