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# PROVE THIS.... THIS IS A BIT HARDER

Note by Sayan Chaudhuri
4 years, 3 months ago

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Hint : Induction. :) · 4 years, 3 months ago

As Zi Song has pointed out, direct induction is one of the easiest ways to go. You can also recognize that $a_{n} = a_{n-1} \times \left(10^{2 \cdot 3^n} + 10^{3^n} + 1 \right) = a_{n-1} \times \left( \left(10^{3^n} \right)^2 + 10^{3^n} + 1\right)$ It should be easy to show that $$3$$ divides $$\left( \left(10^{3^n} \right)^2 + 10^{3^n} + 1\right)$$. Hence, you can now conclude that $$3a_{n-1}$$ divides $$a_n$$. · 4 years, 3 months ago

I came to the same solution as that of Marvis N. Question's not difficult, just presence of mind is needed! · 4 years, 3 months ago

this type of questions i often come across · 4 years, 3 months ago

but i think solution may be somehow critical · 4 years, 3 months ago