Question

if sum of first p terms of an A.P. is ap^2+bq, find its common difference

Answer 1

Sp = ap^2+bq---(1)
S1 =a+bq
S2 = a*2^2 +bq=4a+bq
S3 = a*3^2+bq=9a+bq

a1=S1= a+bq
a2= S2 =a2-a1 = 4a+bq-(a+bq) =4a+bq-a-bq=3a

common difference =d = a2-a1= 3a-(a+bq)
=3a-a-bq
=2a-bq

Answer 2

given sp=ap^2+bq
so its nth term will be given by
an=sn-sn-1
here
a1=s1-s0
a1=a+bq

similarly
a2=s2-s1
a2=a×2×2+bq-(a+bq)
a2=4a+bq-a-bq
a2=3a
so common difference
d=a2-a1
3a-(a+bq)
2a-bq is the common difference