∀ n ϵ N , P ( n ) : 4.7 n + 20 is divisible by 24. State true or false. Options: A . &nbsp True B . &nbsp False C . &nbsp 17 D . &nbsp 24

Answer: Option A : A P ( n ) : 4.7 n + 20 i s d i v i s i b l e b y 24 P ( 1 ) : 28 + 20 = 48 → d i v i s i b l e b y 24 P ( k ) : 4.7 k + 20 i s d i v i s i b l e b y 24 → a s s u m e t r u e ⇒ 4.7 k + 20 = 24 q N o w , 4.7 k + 1 + 20 = 7. ( 4.7 k ) + 20 = 6.4.7 k + ( 4.7 k + 20 ) = 24.7 k + 24 q P ( k + 1 ) i s t r u e ∴ P ( n ) i s t r u e 4.7 n + 20 i s d i v i s i b l e b y 24

∀ n ϵ N, P(n):4.7n+20 is divisible by 24. Sta

Discussion Forum : Principle Of Mathematical Induction

Question -

$\forall n ϵ N, P (n) : {4.7}^{n} + 20$ is divisible by 24. State true or false.

Options:

A . True

B . False

C . 17

D . 24

Answer: Option A
:
A

$P (n) : {4.7}^{n} + 20 i s d i v i s i b l e b y 24$
$P (1) : 28 + 20 = 48 \to d i v i s i b l e b y 24 P (k) : {4.7}^{k} + 20 i s d i v i s i b l e b y 24 \to a s s u m e t r u e \Rightarrow {4.7}^{k} + 20 = 24 q N o w, {4.7}^{k + 1} + 20 = 7. ({4.7}^{k}) + 20 = {6.4.7}^{k} + ({4.7}^{k} + 20) = {24.7}^{k} + 24 q P (k + 1) i s t r u e$
$∴ P (n) i s t r u e {4.7}^{n} + 20 i s d i v i s i b l e b y 24$

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Discussion Forum : Principle Of Mathematical Induction

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