Pythagoras Would Be Proud

Level 2

Consider the sequence $5,12,35,612,93635,\ldots$

What comes next?

4382766512 4383756612 4383766611 4383756514 4383756613

1 solution

Michael Mendrin
May 12, 2016

Follow the pattern

${ 5 }^{ 2 }+{ 12 }^{ 2 }={ \left( 12+1 \right) }^{ 2 }$
${ 12 }^{ 2 }+{ 35 }^{ 2 }={ \left( 35+2 \right) }^{ 2 }$
${ 35 }^{ 2 }+{ 612 }^{ 2 }={ \left( 612+1 \right) }^{ 2 }$
${ 612 }^{ 2 }+{ 93635 }^{ 2 }={ \left( 93635+2 \right) }^{ 2 }$
${ 93635 }^{ 2 }+{ 4383756612 }^{ 2 }={ \left( 4383756612+1 \right) }^{ 2 }$
${ 4383756612 }^{ 2 }+{ 4804330508313429635 }^{ 2 }={ \left( 4804330508313429635+2 \right) }^{ 2 }$

etc.

Is it a coincidence that

$\forall i \in \mathbb{N}, a_{i+2}$ ends in $a_i$

(where the sequence is just $a_i$ )

Wen Z - 4 years, 7 months ago

Pythagoras Would Be Proud

1 solution

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