<s>[INST] Find a pattern and prove it by mathematical induction: $1 = 1$
$3 + 5 = 8$
$7 + 9 + 11 = 27$
$13 + 15 + 17 + 19 = 64$
Etc...

I am having trouble seeing a pattern with this, I know it is relatable with Fibonacci Numbers but I am having trouble grasping this topic
 [/INST] Hint: $$ [1+2\cdot( 1+ 2+\ldots +(n-1))]\cdot n + 2(1+2+\ldots +(n-1)) $$
You can then use Gauss formula for $1+ 2+\ldots +(n-1)$.
As in $n=4$:
$$13 + (13 + 2) + (13 +4) + (13 +6) = 13\cdot 4 + 2\cdot(1+2+3)$$
and 
$$ 13 = 1 + 2\cdot( 1+  2 + 3)$$
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