II picked the following problem from Professor Titu Andreescu’s literature).
Let k be an odd positive integer. Prove that
for all positive integers n.
Proof:
We consider two cases:
Case I: In the first case, we assume that n is odd and write . Then,
. We have
, which, in turn equals,
Since k is odd, is a factor of
. Hence,
divides
for
. Consequently,
divides
. Likewise, we have
which is equal to
Hence, divides
for
. Consequently,
divides
. We have shown that each of
and
divides
. Since
, we conclude that
divides
.
Case II: In the second case, we assume that n is even. The proof is similar to that of the first case. We leave it to the reader.
—-Nalin Pihwa
Related post: http://wp.me/p4LQy6-192
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