demontrer par recurrence que pour tout n>0 appartenent a l ensemble des entier naturel 3^(2n−2^(n ) ) est multiple de 7


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Question Number 176070 by doline last updated on 11/Sep/22

$d e m o n t r e r p a r r e c u r r e n c e q u e p o u r t o u t n > 0 a p p a r t e n e n t a l e n s e m b l e d e s e n t i e r n a t u r e l 3^{2 n - 2^{n}} e s t m u l t i p l e d e 7$
Commented bysom(math1967) last updated on 11/Sep/22

$3^{2 n} - 2^{n}$ $p (1) = 3^{2} - 2^{1} = 7$ $l e t t r u e f o r p (m)$ $∴ 3^{2 m} - 2^{m} = 7 k$ $3^{2 m} = 7 k + 2^{m}$ $p (m + 1) = 3^{2 m + 2} - 2^{m + 1}$ $= 3^{2 m} \times 9 - 2^{m + 1}$ $= (7 k + 2^{m}) \times 9 - 2^{m + 1}$ $= 63 k + 2^{m} \times 9 - 2^{m + 1}$ $= 63 k + 2^{m} (9 - 2)$ $= 7 (9 k + 2^{m})$ $∴ t r u e f o r p (m + 1)$ $∴ 3^{2 n} - 2^{n} m u l t i p l e d e 7 n \in N$
Commented bysom(math1967) last updated on 11/Sep/22

$3^{2 n - 2^{n}} o r 3^{2 n} - 2^{n} ?$
Commented bydoline last updated on 11/Sep/22

$m u l t i p l e d e 7$
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