Question

# If the sum of first n terms of a sequence is of the form An2 + Bw, where A and B arc constants (independent of n). Show that the sequence is an A.P. Is the converse true? Justify your answer.

Here,                                                                     ...(i)
Replacing n by (n - 1) in (i), we get

=                                            ...(ii)

=

Replacing n by (n - 1) in (iii), we get

Now,

which is independent of n

∴       The sequence is an A.P.
Yes, the converse is true.
Let a be the first term and d be the common difference.

∴

where  and  are fixed numbers.
Hence, if the sequence is an A.P. then the sum of first n terms must be of the form