If are in A.P., where for all i, show that

Solution:

It is given that are in the form of AP in which for all i.

To prove that:

Multiply first term by , the second term by and so on i.e., rationalizing each term

Left Hand Side

By using

Left Hand Side

Since are in AP let us assume ‘d’ as its common difference

As a result multiplying by

On putting these values in the Left Hand Side, we get

In the above equation multiply and divide by

The term of the AP is

In which the is last term and
is first term

As a result

Now, substitute in Left Hand Side

As a result hence proved.