Please, I have tried some methods of induction but I can't resolve. Sorry for my english.
I cannot complete to prove. I haved factoring, dividing, adding new terms but i cannot avance for the second step.
So,
$ 4^{2n}-3^{2n}-7 = 84t $
For induction, I have to prove this,
$ 4^{2(k+1)}-3^{2(k+1)}-7 = 84p $
$\endgroup$ 34 Answers
$\begingroup$Basis
$16-9-7=0$, which is divisible by 84.
Induction hypothesis
$84\Big|4^{2k}-3^{2k}-7$
Inductive step
$4^{2k+2}-3^{2k+2}-7$
$=16×4^{2k}-9×3^{2k}-7$
$=16(4^{2k}-3^{2k}-7)+7(3^{2k}+15)$
Now, the left bracket is divisible by 84 by hypothesis and right bracket is clearly divisible by 7×3, to prove that it is also divisible by 4, we write,
$3^{2k}+15=9^k-1^k+16$
Now, $(9-1)|9^k-1^k$ and also 8|16. So, it is divisible by 8, hence by 4.
Hope it helps
$\endgroup$ $\begingroup$If $f(n)=4^{2n}-3^{2n}-7=16^n-9^n-7$
$$f(m+1)-9f(m)=16^m(16-9)-7(1-9)=7(16^m+8)=7(16^m-16+24)$$
will be divisible by $84$
if $16^m-16=16(16^{m-1}-1)$ is divisible by $12$
if $16^{m-1}-1$ is divisible by $3$ which is true as $16^{m-1}-1$ is divisible by $16-1$ for $m\ge1$
$\implies84|f(m+1)\iff84|9f(m)$
Now establish the base case $n=1$
$\endgroup$ $\begingroup$Without induction:
$$F(n)=16^n-9^n-7=16(16^{n-1}-1)-9(9^{n-1}-1)$$
Clearly $F(n)$ is divisible by $3$ as $3|15|(16^{n-1}-1)$ for $n\ge1$
As $9^{n-1}-1$ is divisible by $9-1$ for $n\ge1,8|F(n)$
As $16^n-9^n$ is divisible by $16-9,7|F(n)$
So, $F(n)$ is divisible by lcm$(3,7,8)$
Generalization:
$G(n)=a^n-b^n-(a-b)$
is clearly divisible by $a-b=P$(say)
Again, $G(n)=a(a^{n-1}-1)-b(b^{n-1}-1)$
will divisible by $(a,b-1)=Q$(say) and $(b,a-1)=R$(say)
So, $G(n)$ will divisible by LCM$(P,Q,R)$
$\endgroup$ $\begingroup$Another direct way uses the fact:
- $84 = 3 \cdot 4 \cdot 7$
So, just show divisibility by $\color{blue}{3,4,7}$:
- $4^{2n}-3^{2n}-7 = \color{blue}{(3+1)^{2n}} - 3^{2n} - 7 = \color{blue}{3\cdot k + 1} - 3^{2n} - 7 =\color{blue}{3}\cdot l$
- $4^{2n}-3^{2n}-7 = 4^{2n} - \color{blue}{(4-1)^{2n}} - 7 = 4^{2n} - (\color{blue}{4m + 1}) - 7 =\color{blue}{4}\cdot r$
- $4^{2n}-3^{2n}-7 = \color{blue}{(7-3)^{2n}} - 3^{2n} - 7 = \color{blue}{7s + 3^{2n}} - 3^{2n} - 7 =\color{blue}{7}\cdot t$
