Rewrite as a(n) - 6 a(n-1) + 9 a(n-2), which is a second order linear recurrence.
This has characteristic equation r^2 - 6r + 9 = (r - 3)^2 = 0.
==> r = 3, 3.
So, the general solution is a(n) = (An + B) * 3^n.
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a(0) = 1 ==> 1 = (0 + B) * 3^0 = B.
a(1) = 6 ==> 6 = (A * 1 + 1) * 3^1 ==> A = 1.
Hence, a(n) = (n + 1) * 3^n.
I hope this helps!
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Verified answer
Rewrite as a(n) - 6 a(n-1) + 9 a(n-2), which is a second order linear recurrence.
This has characteristic equation r^2 - 6r + 9 = (r - 3)^2 = 0.
==> r = 3, 3.
So, the general solution is a(n) = (An + B) * 3^n.
----------------
a(0) = 1 ==> 1 = (0 + B) * 3^0 = B.
a(1) = 6 ==> 6 = (A * 1 + 1) * 3^1 ==> A = 1.
Hence, a(n) = (n + 1) * 3^n.
I hope this helps!