Question : If the sum of the first p terms of an AP is ap2+bp, find its common difference.
Doubt by Pushkar
Solution :
Sp = ap2+bp (Given)
Putting p=1
S1=a(1)2+b(1)
S1=a(1)+b
S1=a+b
a1=a+b — (1)
S1=a(1)+b
S1=a+b
a1=a+b — (1)
Putting p=2
S2=a(2)2+b(2)
S2=a(4)+2b
S2=4a+2b
a1+a2=4a+2b
a+b+a2=4a+2b [From eq (1)]
a2=4a+2b-a-b
S2=a(2)2+b(2)
S2=a(4)+2b
S2=4a+2b
a1+a2=4a+2b
a+b+a2=4a+2b [From eq (1)]
a2=4a+2b-a-b
a2=3a+b — (2)
Common difference (d)
d = a2-a1
Common difference (d)
d = a2-a1
d = 3a+b-(a+b)
d = 3a+b-a-b
d = 3a+b-a-b
d = 2a
Hence, the common difference is 2a