Let a 1 , a 2 , … , a 10 be an A . P . with common difference - 3 and b 1 ,   b 2 , … , b…

$C_k=a_k+b_k$

where $a_k=a_1+(k-1)(-3)$ and $b_k=b_1{2}^{k-1}$

$C_k=a_1+(k-1)(-3)+b_1{2}^{k-1}$

$\sum _{k=1}^{10}{c}_k=\sum _{k=1}^{10}{a}_k+\sum _{k=1}^{10}{b}_k$

$C_2=a_1-3+b_1\left(2\right)=12 \left(1\right)....$

$C_3=a_1-6+4{ b}_1=13 \left(2\right)......$

by solving (1) & (2)
$2b_1-3=1\Rightarrow b_1=2$ and $a_1=11$

$\sum _{\mathrm{k}=1}^{10}{\mathrm{c}}_{\mathrm{k}}=\frac{10}{2}\left(22+9\left(-3\right)\right)+\frac{2\left(2^{10}-1\right)}{2-1}$

$=-25+2046=2021$