# Mathematical Induction :)

• Jun 26th 2012, 05:09 AM
Doll0555
Mathematical Induction :)
solve using mathematical induction: 2 times 34^n -3 times 23^n +1 is divisible by 726 for all positive integers n.
• Jun 26th 2012, 06:16 AM
emakarov
Re: Mathematical Induction :)
If $\displaystyle 2\cdot34^n - 3\cdot23^n + 1 = 726k$, then $\displaystyle 68\cdot34^n-102\cdot23^n+34=726k'$, i.e., $\displaystyle 68\cdot34^n = 102\cdot23^n-34+726k'$. Therefore,

$\displaystyle 2\cdot34^{n+1} - 3\cdot23^{n+1} + 1 = 68\cdot34^n - 69\cdot23^n + 1 =$
$\displaystyle 33\cdot23^n-33+726k' = 33(23^n-1) + 726k'$.

So, it is sufficient to prove that $\displaystyle 23^n-1$ is divisible by 22.
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