Answer
a. $0$
b. $7$
c. $8$
d. $-40$
Work Step by Step
Given $\Sigma_{k=1}^{20} a_k=0$ and $\Sigma_{k=1}^{20} b_k=7$, we have:
a. $\Sigma_{k=1}^{20} 3a_k=3\Sigma_{k=1}^{20} a_k=0$
b. $\Sigma_{k=1}^{20} (a_k+b_k)=\Sigma_{k=1}^{20} a_k+\Sigma_{k=1}^{20} b_k=7$
c. $\Sigma_{k=1}^{20} (\frac{1}{2}-\frac{2b_k}{7})=\Sigma_{k=1}^{20} (\frac{1}{2})-\frac{2}{7}\Sigma_{k=1}^{20} b_k=\frac{1}{2}\times20-\frac{2}{7}\times7=8$
d. $\Sigma_{k=1}^{20} (a_k-2)=\Sigma_{k=1}^{20} a_k-=\Sigma_{k=1}^{20}2=0-2\times20=-40$
