For what integers n is 3^(2n+1) + 2^(n+2) divisible by 7?
Can anybody help me?
try using induction
$\displaystyle n=1$
$\displaystyle 3^{2n+1}+2^{n+2}= 3^3 + 2^3 = 27+8 = 35 = 7\cdot 5 $ so it's divisible by 7
$\displaystyle n=2$
$\displaystyle 3^{2n+1}+2^{n+2}= 3^5 + 2^4 = 243+16 = 259 = 7 \cdot 37 $
and so on...
then assume that is true for $\displaystyle n=k$
and if you assume that's true then show that is true and for $\displaystyle n=k+1$ and if you show that... it will mean that it's true for any natural number
hm...
depends, how did you show that ? if not problem post it then we will see is that it
P.S. $\displaystyle 0 \in \mathbb{N} $ that's not true !!!
and induction is based (actually that induction comes from definition of natural numbers) on natural numbers
But it's mentioned integers in the question...So 0 also must be included??
My Proof...
first the base case.
n = 0
3^1 +2^2 = 3+4 = 7
now the more general case.
let k = n. assume 3^(2*k+1) +2^(k+2) is divible by 7.
prove 3^(2*(k+1)+1) +2^((k+1)+2) is divisable by 7.
3^(2*k +3) +2^(k+3)
3^2 *3^(2*k +1) +2* 2^(k+2)
9* 3^(2*k +1) +2* 2^(k+2).
so, based on our assumption, we know 3^(2*k+1) +2^(k+2) is divisable by 7.
we also know based on our formula, that the power for 3 will always increase by 2, and the power for 2 will always increase by 1, for subequent k's. therefore 9* 3^(2*k+1) +2* 2^(k+1) will also be divisble by 7
if one is to use mathematical induction, one must know that :
set of natural numbers $\displaystyle \mathbb{N}$ is subset of set $\displaystyle \mathbb{R}$ with "features" :
1° (1) is natural number, $\displaystyle 1\in \mathbb{N}$
2° any natural number (n) have his follower n'=n+1 which is also natural number $\displaystyle n\in \mathbb{N} \Rightarrow (n+1)\in \mathbb{N}$
3° (1) is not follower of any naturan number
4° if m'=n' than m=n, meaning every natural number is follower of one natural number (and onley one)
5° if $\displaystyle M\subset N $ and if in M apply properties 1° and 2° than M=N
axiom 5 is known as the principle of induction
and so on...
Edit : sorry for this... i was going somewhere else with this.
anyway can't do you harm to know this ( because this is sub forum "university math" so i thought that you would, and should know this )
Yes you should include 0 because the question asks about integers. This poses no problem with respect to using induction.
I don't see how you justified your conclusion at the end. I would rewrite
9* 3^(2*k +1) +2* 2^(k+2)
as
7 * 3^(2k + 1) + 2 * [3^(2k + 1) + 2^(k + 2)]
See?
that is it
$\displaystyle A = 3^{2k+1} + 2^{k+2} $
$\displaystyle 7\cdot 3^{2k+1} + 2\cdot A $
that's it ... because
$\displaystyle 7\cdot 3^{2k+1}$ is divided by 7 ..
and
$\displaystyle 2\cdot A $ is also divided by 7 because you assumed that n=k is true
Hello, yuud!
Here is a sneaky method . . . if you know modulo arithmetic.
$\displaystyle \text{For what integers }n\text{ is }\,3^{2n+1} + 2^{n+2}\,\text{ divisible by 7?}$
$\displaystyle 3^{2n+1} + 2^{n+2} \;\;=\;\left(3^3\right)^\frac{2n+1}{3}} + \left(2^3\right)^{\frac{n+2}{3}} $
. . . . . . . . . . .$\displaystyle =\;(27)^{\frac{2n+1}{3}} + (8)^{\frac{n+2}{2}} $
. . . . . . . . . . .$\displaystyle \equiv\; (\text{-}1)^{\frac{2n+1}{3}} + (1)^{\frac{n+2}{2}}\text{ (mod 7)}$
. . . . . . . . . . .$\displaystyle \equiv\;\left(\sqrt[3]{\text{-}1}\right)^{2n+1} + \left(\sqrt[3]{1}\right)^{n+2}\text{ (mod 7)}$
. . . . . . . . . . .$\displaystyle \equiv\; (\text{-}1)^{2n+1} + (1)^{n+2}\text{ (mod 7)} $
. . . . . . . . . . .$\displaystyle \equiv\; -1 + 1\text{ (mod 7)}$
$\displaystyle 3^{2n+1} + 2^{2n+1} \;\equiv\;0 \text{ (mod 7)}$
It is divisible by 7 for all positive intergers $\displaystyle n$.
I'm not quite familar with the extenstion of operations for modular arithmetic. Maybe a simpler way is: $\displaystyle 3^{2n+1}+2^{n+2}=3\cdot 9^n+4\cdot 2^n $; since $\displaystyle 9\equiv 2 \bmod 7$ and $\displaystyle 4\equiv -3 \bmod 7$ it follows that $\displaystyle 3^{2n+1}+2^{n+2}\equiv 3\cdot 2^n-3\cdot 2^n=0\bmod 7$. Hence disibility by 7 holds for all nonnegative integers $\displaystyle n $.