prove-by-induction-that-4-n-3-n-2-is-a-multiple-of-3-n-Z-

Question Number 73537 by Rio Michael last updated on 13/Nov/19

p r o v e b y i n d u c t i o n t h a t 4^{n} + 3^{n} + 2 i s a m u l t i p l e o f 3

\forall n Z^{+}

Answered by mind is power last updated on 13/Nov/19

n = 1 t r u e

4 + 3 + 2 = 9

s u p p o s e \forall n \in N - {0} u_{n} = 4^{n} + 3^{n} + 2 i s m u l t i p l e o f 3

s h o w 3 ∣ u_{n + 1}

u_{n + 1} = 4 . 4^{n} + 3^{n} * 3 + 2

= 3 . 4^{n} + 4^{n} + 3^{n} + 2 + 2 . 3^{n}

= 4^{n} + 3^{n} + 2 + 3 . 4^{n} + 3^{n} . 2

= u_{n} + 3 (4^{n} + 2 . 3^{n - 1})

s i n c e n ⩾ 1 \Rightarrow 3^{n - 1} \in N \Rightarrow 3 ∣ 3 . (4^{n} + 2 . 3^{n - 1}) b y h y p o t h e s e 3 ∣ U_{n} \Rightarrow

3 ∣ u_{n + 1}

Commented by Rio Michael last updated on 13/Nov/19

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