If the sum of p terms of an A.P. is q and the sum of q terms is p, show that the sum of p + q terms is – (p + q). Also, find the sum of first p – q terms (p > q).

Solution:

An AP’s sum of n terms is given by

Where the first term is ‘a’ and the common difference is ‘d’.

It is given that and

It is known that

Now rearrange

Again we have

When we rearrange we obtain

Subtract eq.(i) from eq.(ii) i.e. eq.(ii) – eq.(i)

Subtract eq.(i) from eq.(ii) i.e. eq.(ii) – eq.(i)

When we simplify we obtain

Using the formula we obtain

Compute and simplify now,

We need to show that

Above eq. can also be re-written as

Now using the

From eq.(iii) substitute ‘d’

We now need to find the sum of terms that is

The above eq. can also be re-written as

The above eq. can also be re-written as

Now using and

On substituting the value of we obtain

From eq.(iii) substitute ‘d’

Simplify and compute now

As a result the sum of terms is