Three straight lines cross through a circle. What is the maximum number of non-overlapping sectors in which the circle is divided?

In this case a "sector" is just a piece of the divided circle.
Granted: the wording is sloppy since the question is talking about a circle.

In general if we want to divide the circle with the nth line we get exactly as many *new pieces* as the nth line is divided into by the previous n-1 lines i.e. the nth line produces n new pieces.
Hence, if f(n) is the number of pieces we get from n lines we have from the explanation above:

f(n) = f(n-1) + n

with f(0)=1

f(1) = f(0) + 1 = 2
f(2) = f(1) + 2 = 4
f(3) = f(2) + 3 = 7 <----

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In general we see that for n lines
f(n) = n + (n-1) + (n-2) + ... +1 + 1
= n(n+1)/2 + 1

f(3) = 3*4/2 + 1 = 6+1 = 7

A quick Google of the word "sector" produced this pair of definitions:

1. an area or portion that is distinct from others.
2. the plane figure enclosed by two radii of a circle or ellipse and the arc between them.

Definition 1 is the general meaning of the word. And in that sense, the book answer is correct.

Definition 2 is a use of the word which is particular to geometry of circles. And yes, it might have been better if they hadn't used "sector" to mean something else when they were talking about geometry of circles.