Let Fn be the nth Fibonacci number. Deduce these properties WITHOUT .pdf

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Let Fn be the nth Fibonacci number. Deduce these properties WITHOUT using the method of mathematical induction. Fn=2Fn-2+Fn-3 Solution we know fibonacci numbers are related as Fn = Fn-1+Fn-2 thus Fn-1 = Fn-2 + Fn-3 thus Fn = Fn-1 + Fn-2 = Fn-2 + Fn-3 +Fn-2 = 2 Fn-2 + Fn-3 thus ans.

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Let R be the ring of 2 × 2 matrices of the form a b 0 0 where a, b ? R. Show that although R is a ring that has no identity, we can find a subring S of R with an identity. Solution Take S to be the set of matrices of the form a 0 0 0 where a is in R. It is completely obvious that S is closed under both multiplication and addition. [In fact S is isomorphic to R]. Now the (multiplicative) identity in S is: 1 0 0 0 again this is completely obvious (what happens when you multiply something in S by the matrix on the left or right?). I hope this helps, if you need more details write back..

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