Explanation
We are given the following series:
[tex]12+4-4-12-20-...[/tex]
We are required to determine the sum of the first 14 terms of the given series.
This is achieved thus:
We know that the sum of n terms of a series is given as:
[tex]\begin{gathered} S_n=\frac{n}{2}[2a+(n-1)d] \\ where \\ a=first\text{ term} \\ d=common\text{ difference} \\ n=number\text{ of terms} \end{gathered}[/tex]
Therefore, we have:
[tex]\begin{gathered} S_n=\frac{n}{2}[2a+(n-1)d] \\ where \\ a=12 \\ d=4-12=-8 \\ n=14 \\ \\ \therefore S_{14}=\frac{14}{2}[2\cdot12+(14-1)-8] \\ S_{14}=7[24+(13)-8] \\ S_{14}=7(24-104) \\ S_{14}=7\cdot-80 \\ S_{14}=-560 \end{gathered}[/tex]
Hence, the answer is:
[tex]S_{14}=-560[/tex]
