1964 AHSME Problems/Problem 31: Difference between revisions

Revision as of 16:54, 5 March 2014

Let $f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n.$

Then $f(n+1)-f(n-1)$ , expressed in terms of $f(n)$ , equals:

$\textbf{(A) }\frac{1}{2}f(n)\qquad\textbf{(B) }f(n)\qquad\textbf{(C) }2f(n)+1\qquad\textbf{(D) }f^2(n)\qquad \textbf{(E) }$ $\frac{1}{2}(f^2(n)-1)$

Revision as of 16:50, 5 March 2014 view source Bobthesmartypants (talk \| contribs) editor 292 edits No edit summary ← Older edit		Revision as of 16:54, 5 March 2014 view source Bobthesmartypants (talk \| contribs) editor 292 edits No edit summary Newer edit →
Line 1:		Line 1:
			==Problem==

	Let <cmath>f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n.</cmath>		Let <cmath>f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n.</cmath>