To find maximum number of regions in to which ‘n’ straight
lines can divide a plane:

“

Let us go case by case starting with one straight line.

**Maximum number of regions can be achieved when the lines are drawn such that no two are parallel and no three are concurrent**”Let us go case by case starting with one straight line.

One straight line can divide a plane into

**2**.
Two straight lines can divide a plane into

**4**.
Three straight lines can divide a plane into

**7**.
observe the sequence: 2, 4, 7,....

If we observe, it is a special sequence which can be written
as

1+1, 3+1, 6+1,......

= (1)+1, (1+2)+1, (1+2+3)+1,.....

= (1)+1, (1+2)+1, (1+2+3)+1,.....

= ∑1 + 1, ∑2 + 1, ∑3 + 1,......

So, we can generalise like this:

**Maximum number of regions that ‘n’ straight lines can divide a plane = ∑n + 1 = n(n+1)/2 + 1**
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