Let us take example of an auditorium and find the sum of AP.

Let l denotes the number of seats in the last of n^th row.

Now, there are n rows. The front row has a seats. The last row has l seats.

Let S denotes the total number of seats.

Now, copy the diagram, turmit upside and place it next to the given diagram.

Two copies of the auditorium produces a figure consisting of n rows each having a + l seats.

Id S is the sum of the seats in the auditorium,

Then 2S = n(a + l)

=> S = (n/2) * (a + l) ..........1

If the last row is the n^th row, then

l = a + (n - 1)d

From equation 1, we get

=> S = (n/2) * {a + a + (n - 1)d}

=> S = (n/2) * {2a + (n - 1)d}

This gives the sum of n terms of the an AP.