Prove by Math induction that for any positive integer n,
(7n+1)6^n + (-1)^(n+1)
is divisible by 49.
Thanks
Simon YAU
Difficult Math Induction
2012-07-29 3:31 am
回答 (2)
2012-07-29 4:55 am
2012-07-29 5:21 am
唔做代n=1個step....
(7k+1)6^k + (-1)^(k+1)=49M
(7k+1+7)6^(k+1) + (-1)^(k+1+1)
【(7k+1+7)6^k 】(6) + (-1)【49M- (7k+1)6^k】
【(7k+1+7)6^k 】(6) - 49M + (7k+1)6^k
6^k【42k+48+7k+1】 - 49M
6^k(49k+49) -49M
49【6^k(k+1) -M 】
so,P(n+1) is also true
by MI, it is true for any positive integer n
(7k+1)6^k + (-1)^(k+1)=49M
(7k+1+7)6^(k+1) + (-1)^(k+1+1)
【(7k+1+7)6^k 】(6) + (-1)【49M- (7k+1)6^k】
【(7k+1+7)6^k 】(6) - 49M + (7k+1)6^k
6^k【42k+48+7k+1】 - 49M
6^k(49k+49) -49M
49【6^k(k+1) -M 】
so,P(n+1) is also true
by MI, it is true for any positive integer n
參考: me
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