. Consider the sequence defined recursively as a0 = 5, a1 = 16 and ak = 7ak−1 − 10ak−2 for all integers k ≥ 2. Prove that an = 3 · 2 n + 2 · 5 n for each integer n ≥ 0
We are going to prove by strong induction, that the given
recursive sequence satisfies
.
For n=0,
and for n=1,
.
Now suppose, the given recursive sequence satisfies the answer
for all
, that is,
for all
for
some
. Then,
.
Hence, we have proved by induction that the given recursive
sequence satisfies
.
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