<s>[INST] Proof by induction for a sequence, given $a_1$ and $a_2$ Question:  prove by induction that this statement holds.
Define  the sequence $(a_{k}),  k∈\mathbb N ,$
by $a_1 = 2, a_2 = 5 $
and
$a_{k+2} = 5a_{k+1} − 6a_{k} $ for all $k≥1$. Then $a_{k} =2^{n−1} +3^{n-1}$ for all $n≥1 $.
What I did was find the base case -- $ P(1)=2 $ -- holds
Then made an assumption that $P(k)$ holds.  Thus we can assume $P(k+1)$ and I got that
$= 5×2^{k−1} +5×3^{k−1} −3×2^{k−1} −2×3^{k−1}.$
Where do I go from here?
 [/INST] Say $P(n)$ is $a_n=2^{n-1}+3^{n-1}$.
Base:  $P(1):  a_1=2=2^0+3^0$, $P(2):  a_2=5=2^1+3^1$.
Now assuming $P(k)$ and $P(k+1)$, we have
$P(k+2):  a_{k+2}=5a_{k+1}-6a_k=5(2^k+3^k)-6(2^{k-1}+3^{k-1})$
$=(5-3)2^{k}+(5-2)3^{k}=2^{k+1}+3^{k+1}.$
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