I've seen the property   $$f_{n+1} f_{n−1} = f_n^2 + (−1)^n, n ≥ 2.$$   for Fibonacci numbers at Abstract Algebra book of Thomas W. Judson. I've tried it for a few Fibonacci number, and I've really liked how the pattern goes. I've tried to prove it by induction, however I failed to do so. I would appreciate, if someone could prove and explain to me.   I would also like to state that, I'm not a math major. Therefore I would be really glad if you prove it as simple as possible.   Thanks.
How do I see that $$f_{n+1}f_{n-1} = f_n^2 + (-1)^n$$ for $n \ge 2$, where $f_1 = 1$, $f_2 = 1$, and $f_{n+2} = f_{n+1} + f_n$ for $n \in \mathbb{N}$?
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It turns out that SharePoint decided it would be a good idea to inexplicably add a few lines of:  nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;  nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;   between my   in my DevHome.aspx (I'm using Angular for this particular project). So, why does Sharepoint do this, and how do I prevent it? This is not the first time SharePoint has randomly added an nbsp to my page.  Thanks in advance. Number theorems with chapter number I use declaretheoremstyle to define a theorem environment, and print a list of theorems in the end of my document:  \documentclass[12pt]{book} \usepackage[english]{babel} \usepackage[margin=2cm]{geometry}% just for the example \usepackage{xcolor} \usepackage{amsthm} \usepackage{thmtools} \usepackage{mdframed} \usepackage{etoolbox}  \declaretheoremstyle[   spaceabove=6pt, spacebelow=6pt,   headfont=\normalfont\bfseries,   postheadspace=1em,   notefont=\bfseries,   notebraces={(}{)},   bodyfont=\itshape,   shaded={bgcolor=yellow!20} ]{thmstyle}  \declaretheorem[style=thmstyle,name=Theorem]{theorem}  \begin{document}  \begin{theorem}[Somebody] ... \end{theorem}  \begin{theorem} ... \end{theorem}  \renewcommand{\listtheoremname}{List of Theorems} \addcontentsline{toc}{section}{\listtheoremname} \listoftheorems[ignoreall,show=theorem]  \end{document}   The theorems are numbered from the beginning to the end of the book, and from 1:    However, I would like to number the theorems with chapter number (like Figures or Tables), for instance, 1.1 (the first theorem in Chapter 1), 1.2 (the first theorem in Chapter 2)...  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