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Category: Differentiation

discrete-mathematics-prove-that-n-1-1-F-2n-1-1-5-5-2-F-n-fibonacci-sequence-

Question Number 142723 by mnjuly1970 last updated on 04/Jun/21 $$\:\:\:\:\:\:\:\:…….\:{discrete}\:\:…..\:\:{mathematics}……. \\ $$$$\:\:\:\:{prove}\:{that}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{F}_{\mathrm{2}{n}+\mathrm{1}} −\mathrm{1}}\overset{?} {=}\frac{\mathrm{5}−\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:{F}_{{n}} \:::\:{fibonacci}\:\:{sequence}… \\ $$ Answered by…

A-geometric-sequence-with-n-terms-a-1-a-2-a-3-a-n-which-has-a-1-a-n-3-If-the-product-of-all-n-terms-a-1-a-2-a-3-a-n-59049-Determine-the-value-of-n-

Question Number 11633 by Joel576 last updated on 29/Mar/17 $$\mathrm{A}\:\mathrm{geometric}\:\mathrm{sequence}\:\mathrm{with}\:{n}\:\mathrm{terms}\: \\ $$$${a}_{\mathrm{1}} ,\:{a}_{\mathrm{2}} ,\:{a}_{\mathrm{3}} ,\:…,\:{a}_{{n}} \:\mathrm{which}\:\mathrm{has}\:{a}_{\mathrm{1}} \:.\:{a}_{{n}} \:=\:\mathrm{3} \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{all}\:{n}\:\mathrm{terms}\:=\:{a}_{\mathrm{1}} {a}_{\mathrm{2}} {a}_{\mathrm{3}} …{a}_{{n}} =\:\mathrm{59049} \\…

Prove-that-n-N-k-1-n-1-sin-kpi-2n-k-1-n-1-cos-kpi-2n-

Question Number 142698 by Willson last updated on 04/Jun/21 $$\mathrm{Prove}\:\mathrm{that}\:\forall\mathrm{n}\in\mathbb{N}^{\ast} \\ $$$$\:\:\:\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}−\mathrm{1}} {\prod}}\mathrm{sin}\left(\frac{\mathrm{k}\pi}{\mathrm{2n}}\right)\:=\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}−\mathrm{1}} {\prod}}\mathrm{cos}\left(\frac{\mathrm{k}\pi}{\mathrm{2n}}\right) \\ $$ Answered by Ar Brandon last updated on…