- n - N - P n - 4-7 n -20 is divisible by 24 - State true or false-A- TrueB- False

The correct option is A True P(n): 4.7^n+20 is divisible by 24 P(1): 28+20=48 → divisible by 24 P(k): 4.7^k+20 is divisible by 24 → assume true ⇒ 4.7^k+20 ...

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Byju's Answer

Standard XI

Mathematics

Proof by mathematical induction

∀ n ∈ N , P n...

Question

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

True

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False

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Solution

The correct option is A
True

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

$∴ P (n) i s t r u e {4.7}^{n} + 20 i s d i v i s i b l e b y 24$

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∀ n ϵ N, P(n):4.7n+20 is divisible by 24. State true or false.

The correct option is A True P(n):4.7n+20 is divisible by 24 P(1):28+20=48→divisible by 24P(k):4.7k+20 is divisible by 24→assume true⇒4.7k+20=24qNow, 4.7k+1+20=7.(4.7k)+20=6.4.7k+(4.7k+20)=24.7k+24qP(k+1) is true ∴P(n) is true4.7n+20 is divisible by 24

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