Answer
Arithmetic;
Sum: $\dfrac{27n-n^2}{4}$
Work Step by Step
We are given the sequence:
$\left\{-\dfrac{n}{2}+7\right\}$
Determine the difference of consecutive terms:
$a_{n+1}-a_n=-\dfrac{n+1}{2}+7-\left(-\dfrac{n}{2}+7\right)$
$=-\dfrac{n}{2}-\dfrac{1}{2}+7+\dfrac{n}{2}-7$
$=-\dfrac{1}{2}$
As the difference of consecutive terms is constant, the sequence is arithmetic. Its elements are:
$a_1=-\dfrac{1}{2}+7=\dfrac{13}{2}$
$d=-\dfrac{1}{2}$
Determine the sum of the first $n$ terms:
$S_n=\dfrac{n(2a_1+(n-1)d)}{2}$
$S_n=\dfrac{n\left(2\left(\dfrac{13}{2}\right)+(n-1)\left(-\dfrac{1}{2}\right)\right)}{2}=\dfrac{n(26-n+1)}{4}=\dfrac{n(27-n)}{4}=\dfrac{27n-n^2}{4}$
