if pth term of an ap is q and the q th term is p show that r th term is p+q-r

Dear Student,

 

Let the first term of the AP be A and common difference be D.

 

Now, from the given conditions,

 

A + (p-1)D = q and A + (q-1)D = p

 

Subtracting the two equations, we get

pD - D - qD + D = -(p - q)

=> (p - q)D = -(p - q)

=> D = -1

 

putting D = -1 in the first equation, we get

A = q + (p - 1) = p + q - 1

 

Therefore, the r^th term of the AP is

A + (r - 1)D = p + q - 1 - r + 1 = p + q - 1

Hence, proved.

 

Regards Topperlearning.