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- #1

- Apr 13, 2013

- 3,718

Suppose that we have the recurrence relation $a_k=3^k-a_{k-1}, a_0=1$.

By replacing the terms of the sequence we get that it is equal to $a_k=3^k-3^{k+1}+3^{k+2}-a_{k-3}$.

How do we get that it is equal to $a_k=3^{k}-3^{k-1}+3^{k-2}- \dots+ (-1)^k$ ?

Also, why is the latter equal to $(-1)^k (3^k+3^{k-1}+\dots+1)$ ?