Answer
$-2700$
Work Step by Step
We are given that the sequence $ a_n $ is a geometric sequence and the sequence $ b_n $ is an arithmetic sequence.
Here, we have $ r=-2$ and $ d=-15$
The general formula to find the sum of first n term of a Geometric sequence is given as: $ S_{n}=\dfrac{a_1(1-r^n)}{(1-r)}$
Thus, $ S_{11}=\dfrac{(-5) \times (1-(-2)^{11})}{[1-(-2)]} \\=\dfrac{(-5) \times (1+2^{11})}{[1+2]}=-3415 $
The general formula to find the sum of the first n term of an Arithmetic sequence is given as: $ S'_{n}=\dfrac{n(b_1+b_n)}{2}$
and $ S'_{11}=\dfrac{11(b_1+b_1+(n-1)d)}{2} \\=\dfrac{11(10+10+(11-1) (-15))}{2}=-715$
Now, $ S_{110}+S'_{11}=-3415-715=-2700$
