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Permutations And Combinations
If the sum of first n terms of a sequence is of the form An^{2} + 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
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Determine K, so that K + 2, 4K – 6 and 3K – 2 are three consecutive terms of an A.P.
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