# Relationship between mapping cones and spectral sequences

I'm learning homological algebra using Weibel's book. And I have trouble in exercise 5.4.4:

Let $$f : B \to C$$ be a map of filtered chain complexes. For each $$r \geq 0$$, define a filtration on the mapping cone $$cone(f)$$ by $$F_p cone(f)_n = F_{p-r} B_{n-1} \oplus F_p C_n.$$ Show that $$E^r_p (cone f)$$ is the mapping cone of $$f^r : E^r_p (B) \to E^r_p (C)$$.

I'm not used to deal with mapping cone and spectral sequence, and I'm curious about the approach to this problem.

At first, I tried to represent $$E^r_p (conef)$$ explicitly by the construction process of $$E^r_p$$, but the indexing and calculation were so complicated that I doubted if it was the right way.