document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199565>Question 1199565</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A function f is defined on the set of positive integers by f(1) = 1, f(3) = 3, f(2n) = n, f(4n + 1) = 2f(2n + 1) − f(n), and f(4n + 3) =<BR>\n" );
document.write( "3f(2n + 1) − 2f(n) for all positive integers n. Determine the summation of [f(4n + 1) + f(2n + 1) − f(4n + 3)] if i = 1 and n=10. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #833520 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=833520\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "I'm not sure how to read the summation you/we are supposed to find.  So I'll leave that part of the problem to you; I will only show how to get the function values f(n) for each integer n.<br><BR>\n" );
document.write( "(1) We are given f(1) = 1 and f(3) = 3; for all other integers n, the values of f(n) will be determined recursively from those two given function values.<br><BR>\n" );
document.write( "(2) f(2n) = n<br><BR>\n" );
document.write( "This defines the function values of the function when the function argument is even: f(2) = 1; f(4) = 2; f(6) = 3; etc.<br><BR>\n" );
document.write( "(3) When the function argument is odd, it is either of the form 4n+1 or 4n+3.  The other two pieces of given information tell us how to find the function values when the argument is odd (and not equal to 1 or 3).<br><BR>\n" );
document.write( "The two recursive definitions for these function values are<br><BR>\n" );
document.write( "f(4n + 1) = 2f(2n + 1) − f(n) [1]<BR>\n" );
document.write( "f(4n + 3) = 3f(2n + 1) − 2f(n)[2]<br><BR>\n" );
document.write( "Using these definitions....<br><BR>\n" );
document.write( "f(5): 5 = 4(1)+1, so using [1], f(5) = 2(f(3))-f(1) = 2(3)-1 = 5<BR>\n" );
document.write( "f(7): 7 = 4(1)+3, so using [2], f(7) = 3(f(3))-2(f(1)) = 3(3)-2(1) = 7<BR>\n" );
document.write( "f(9): 9 = 4(2)+1, so using [1], f(9) = 2(f(5))-f(2) = 2(5)-1 = 9<BR>\n" );
document.write( "f(11): 11 = 4(2)+3, so using [2], f(11) = 3(f(5))-2(f(2)) = 3(5)-2 = 13<BR>\n" );
document.write( "f(13): 13 = 4(3)+1, so using [1], f(13) = 2(f(7))-f(3) = 14-3 = 11<BR>\n" );
document.write( "f(15): 15 = 4(3)+3, so using [2], f(15) = 3(f(7))-2(f(3)) = 21-6 = 15<BR>\n" );
document.write( "f(17): 17 = 4(4)+1, so using [1], f(17) = 2(f(9))-f(4) = 18-2 = 16<BR>\n" );
document.write( "f(19): 19 = 4(4)+3, so using [2], f(19) = 3(f(9))-2(f(4)) = 27-4 = 23<BR>\n" );
document.write( "...<br><BR>\n" );
document.write( "I'm not seeing an easy pattern....<br><BR>\n" );
document.write( "And, as I stated earlier, I'm not sure what summation we are supposed to be doing....<br><BR>\n" );
document.write( "So I will end my response here.<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );