If the sum of n terms of an A.P. is ( pn + qn 2 ), where p and q are constants, find the common difference.
The sum of n terms of an A.P. is ( p n + q n 2 ) .
The formula for the sum of n terms in an A.P. is given by,
S n = n 2 [ 2 a + ( n − 1 ) d ]
Substitute the value of S n as ( p n + q n 2 ) in the above expression.
p n + q n 2 = n 2 [ 2 a + ( n − 1 ) d ] p n + q n 2 = n 2 [ 2 a + n d − d ] = n a + d 2 n 2 − d 2 n p n + q n 2 = ( a − d 2 ) n + d 2 n 2
Compare the coefficients on both the sides.
d 2 = q d = 2 q
Thus, the common difference of the A.P. is 2 q .
The sum of
The formula for the sum of
Substitute the value of
Compare the coefficients on both the sides.
Thus, the common difference of the A.P. is
