Answer
$a\displaystyle \cdot\frac{1-a^{9}}{1-a}+55b$
Work Step by Step
Partial sums of
a geometric sequence: $S_{n}=a\displaystyle \cdot\frac{1-r^{n}}{1-r}$,
an arithmetic sequence: $S_{n}=n(\displaystyle \frac{a+a_{n}}{2})$
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$(a+b)+(a^{2}+2b)+(a^{3}+3b)+\cdots+(a^{10}+10b) =$
$=(a+a^{2}+a^{3}+\cdots+a^{10})+(b+2b+3b+\cdots+10b)$
The first sum is a sum of a geometric sequence,
first term: $a$, ratio: $a$.
The second is a sum of an arithmetic sequence,
first term: $b$, common difference: $b.$
$=a\displaystyle \cdot\frac{1-a^{9}}{1-a}+\frac{10}{2}\cdot(b+10b)$
$=a\displaystyle \cdot\frac{1-a^{9}}{1-a}+55b$
