## Precalculus (10th Edition)

$-7592$
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: $S_n=\dfrac{n(a_1+a_n)}{2}$ $7+1-5-11-...-299=\dfrac{52(7-299)}{2}$ $=\dfrac{52\cdot (-292)}{2}$ $=-7592$