I'm having trouble tackling this ghastly exercise.

Let $(a_n)_{n\in\mathbb{N}}:a_1=3,a_2=-1,a_{n+1}=a_{n+1}+4^{2n}a_n+15^n n^{15}$. Prove that $a_n \equiv 3^n \pmod 5$.

I know that every term in this sequence is congruent modulo 5 to the sum of the previous two terms in the sequence:

$$a_{n+2}=a_{n+1}+4^{2n}a_n+15^n n^{15} \equiv a_{n+1} + a_n \pmod 5$$

But I don't know how to find a explicit formula for $a_n$. Is it crucial to do so? [Edit: can I treat $(a_n)$ as equivalent to $(b_n): b_1=3,b_2=-1,b_{n+2}=b_{n+1}+b_n$ for the purposes of this exercise?]

Does this even require induction on $n$?

I'm grasping at straws here.

  • $\begingroup$ Please fix the typos in the definition of your sequence. $\endgroup$ – darij grinberg May 13 at 6:26

You already got $a_{n+2}\equiv a_{n+1}+a_{n}$(mod 5), but $3^{n+2}\equiv 3^{n+1}+3^n$ (mod 5) because $9-3-1=5$. Also $a_1=3^1$ and $a_2=-1\equiv 3^2$ (mod 5) thus we are done.

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    $\begingroup$ I guess you're applying induction on $n$. Your induction hypothesis would be $a_n \equiv 3^n \pmod 5, a_{n+1} \equiv 3^{n+1} \pmod 5$. And you'd be then proving that $a_{n+2} \equiv 3^{n+2} \pmod 5$. But can you always treat the indices of the terms as bases for induction? $\endgroup$ – ydnfmew May 13 at 6:40
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    $\begingroup$ @ydnfmew Yes, that is correct. $\endgroup$ – Peter Foreman May 13 at 6:43

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