Thread: Induction proof

hey all hope ur fine,
i have a homeworks really hard bcz i was absent for 2 weeks(i was sick)
so plz can u help me??

the lesson name: Recurrence

1) n being an entire naturel,proove that:
3^(2n)_ 2^(n) divisible by 7
2)5^(2n)_2^(n) divisible by 23

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proove one polynome p(x)=ax^2+bx that verifie,for all reel x's, p(x+1)_p(x)=x

etablissez by recurrence that p(n) entire east for all n E N

be the term continuation generale Sn=1+2+......+n, n_>1

hey sorry for my english iam learning math in french but this forum da best many thanks

Hello, iceman1!

I can help with the first problem . . .

1) n is a natural number. .Prove that:

(a) 3^{2n} - 2^n is divisible by 7

(b) 5^{2n} - 2^n is divisible by 23

(a) We have: .N .= .(3²)^n - 2^n .= .9^n - 2^n .= .(7 + 2)^n - 2^n

Expand the binomial:
N .= .[7^n + n·7^{n-1}·2 + [n(n-1)/2]·7^{n-2}·2² + . . . + n·7·2^{n-1} + 2^n] - 2^n

And we have:
N .= .7^n + n·7^{n-1}·2 + [n(n-1)/2]·7^{n-2}·2² + ... + n·7·2^{n-1}

Since every term has a factor of 7, N is divisible by 7.



(b) We have: .N .= .(5²)^n - 2^n .= .25^n - 2^n .= .(23 + 2)^n - 2^n

Now proceed as in part (a).