Answer
$-7592$
Work Step by Step
We have to determine the sum:
$S=7+1-5-11-.....-299$
As $1-7=-5-1=-11-(-5)=...=-6$, the sequence is arithmetic.
Determine its first term and the common difference:
$a_1=7$
$d=-6$
Determine the number of terms:
$a_n=a_1+(n-1)d$
$a_n-a_1=(n-1)d$
$n-1=\dfrac{a_n-a_1}{d}$
$n=\dfrac{a_n-a_1}{d}+1$
$n=\dfrac{-299-7}{-6}+1$
$n=52$
The given sum contains $52$ terms, so we have to determine the sum of the first $52$ terms. We use the formula:
$S_n=\dfrac{n(a_1+a_n)}{2}$
$7+1-5-11-...-299=\dfrac{52(7-299)}{2}$
$=\dfrac{52\cdot (-292)}{2}$
$=-7592$
