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https://texercises.com/exercise/linearity-fibonacci-sequences/
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Exercise:
abcliste abc If mathcalF_a_a_... and mathcalF_b_b_... are both Fibonacci sequences this implies that mathcalF_+mathcalF_a_+b_a_+b_... is also a Fibonacci sequence. abc Let mathcalFa_a_... be a Fibonacci sequence and alpha in mathbbR. Define alpha mathcalFalpha a_alpha a_.... Show that alpha mathcalF is also a Fibonacci sequence. abcliste

Solution:
abcliste abc Proof. mathcalF_+mathcalF_c_c_.... We need to show that c_n c_n-+c_n-. Therefore we first write c_n c_n- c_n- a bit differently c_n a_n+b_n c_n- a_n-+b_n- c_n- a_n-+b_n- which then leads to c_n-+c_n- a_n-+b_n-+a_n-+b_n- a_n-+a_n-+b_n-+b_n- a_n+b_n c_n abc We need to check that alpha a_n alpha a_n-+alpha a_n-. But this ist obviously true since a_n a_n-+a_n-. In fact we see that alpha mathcalF_a_a_mathcalF_alpha a_alpha a_ abcliste
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\(\LaTeX\)-Code
Exercise:
abcliste abc If mathcalF_a_a_... and mathcalF_b_b_... are both Fibonacci sequences this implies that mathcalF_+mathcalF_a_+b_a_+b_... is also a Fibonacci sequence. abc Let mathcalFa_a_... be a Fibonacci sequence and alpha in mathbbR. Define alpha mathcalFalpha a_alpha a_.... Show that alpha mathcalF is also a Fibonacci sequence. abcliste

Solution:
abcliste abc Proof. mathcalF_+mathcalF_c_c_.... We need to show that c_n c_n-+c_n-. Therefore we first write c_n c_n- c_n- a bit differently c_n a_n+b_n c_n- a_n-+b_n- c_n- a_n-+b_n- which then leads to c_n-+c_n- a_n-+b_n-+a_n-+b_n- a_n-+a_n-+b_n-+b_n- a_n+b_n c_n abc We need to check that alpha a_n alpha a_n-+alpha a_n-. But this ist obviously true since a_n a_n-+a_n-. In fact we see that alpha mathcalF_a_a_mathcalF_alpha a_alpha a_ abcliste
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Tags
eth, fibonacci, hs22, linalg i, proof
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Difficulty
(2, default)
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Language
ENG (English)
Type
Proof
Creator rk
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