9|((n*4^n+1) - (n+1)*4^n  + 1)

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aruv | High School Teacher | (Level 2) Valedictorian

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P(n):`9| (nxx4^(n+1)-(n+1)xx4^n+1)`

P(n) is statement that 9 divides `(nxx4^(n+1)-(n+1)xx4^n+1).`

Let check if P(n) is true for n=1,

`9|(4^2-8+1)`

`=> 9|9, ` which is true.

Let us assume P(n) is true for n=k i.e.

`p(k): 9|(kxx4^(k+1)-(k+1)xx4^k+1)`

Now prove that P(n) is true for n=k+1, when P(k) is true i.e.

`P(k+1): 9| ((k+1)xx4^(k+2)-(k+2)xx4^(k+1)+1)`

R.H.S.=`(k+1)4^(k+2)-(k+2)4^(k+1)+1`

`=kxx4xx4^(k+1)+4^(k+2)-4(k+1)4^k-4xx4^k+1`

`=4(kxx4^(k+1)-(k+1)4^k+1)+4^(k+2)-4xx4^k-3`

`=4(kxx4^(k+1)-(k+1)4^k+1)+3(4^(k+1)-1)`

Since P(k) is true and `(4^(k+1)-1)`  is always multiple of 3. Therefore P(k+1) is true. Thus P(n) is true for all n.

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