2008 UNCO Math Contest II Problems/Problem 9
(a) Prove that
(b) Prove that
(c) Prove that each term in the following sequence is a perfect square:
(a) We know that is a geometric series, so we can define it explicitly as follows
multiplying both sides by 9 yields our answer.
(b) We have
(c) We say that the nth member of the sequence equals . Expanding yields
Dividing each term separately, we know that the first term will add s and , the second term will add s and , and the third will add , giving
which is exactly what we wanted.
(a) (b) (c)
(a) . Multiply these two binomials and we have reach our answer (remember the formula -- it's like Difference of Cubes)
(b). The original expression is equal to . (Just brute force this out).
(c) Now notice that each term in the sequence is . As seen in part (a), we see that . Follow Solution 1 above.
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