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Mathematics
Problem
Prove by induction that, for
n
∈
Z
+
n \in \mathbb{Z}^+
n
∈
Z
+
, a.
f
(
n
)
=
5
n
+
8
n
+
3
f(n) = 5^n + 8n + 3
f
(
n
)
=
5
n
+
8
n
+
3
is divisible by 4, b.
(
3
−
2
2
1
)
n
=
(
2
n
+
1
−
2
n
2
n
1
−
2
n
)
.
\begin{pmatrix} 3 & -2 \\ 2 & 1 \end{pmatrix}^n = \begin{pmatrix} 2n+1 & -2n \\ 2n & 1-2n \end{pmatrix}.
(
3
2
−
2
1
)
n
=
(
2
n
+
1
2
n
−
2
n
1
−
2
n
)
.
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