Answer
$$0.692580927$$
Work Step by Step
We are given that $\dfrac{s_n+s_{n+1}}{2}=s_n+\dfrac{1}{2} (-1)^{n+2} a_{n+1}$
This gives that $s_n=\dfrac{s_{n+1}}{2}-\dfrac{1}{2} (-1)^{n+2} a_{n+1}$
Thus, $s_{20}= 1-\dfrac{1}{2}+\dfrac{1}{3}-......-\dfrac{1}{20}$
So, $s_{20} \approx 0.66 87714032$
Now, $s_{20}+(\dfrac{1}{2}) (\dfrac{1}{21}) = 0.66 87714032+\dfrac{1}{42}=0.692580927$
