9395 /www/nrich/html/content/id/9395/ 1 2012-09-14T15:10:53 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <br></br> In a sequence of positive integers, every term after the first two terms is the sum of the two previous terms in the sequence.<br></br> <br></br> If the fifth term is $2004$, what is the maximum possible value of the first term?<br></br> <br></br> <br></br> <br></br> <br></br> If you liked this problem, <a href="/1019">here is an NRICH task</a> which challenges you to use similar mathematical ideas.<br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Let the first two terms of the sequence be $a$ and $b$ respectively.<br></br> <br></br> Then the next three terms are $a+b$, $a+2b$, $2a+3b$. So $2a+3b = 2004$.<br></br> <br></br> For $a$ to be as large as possible, we need $b$ to be as small as possible, consistent with both being positive integers.<br></br> <br></br> If $b=1$ then $2a=2001$, but $a$ is an integer, so $b\not=1$.<br></br> <br></br> However, if $b=2$ then $2a=1998$, so the maximum possible value of $a$ is $999$.<br></br> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Patterns and Sequences - Stage 4 Short Problem UKMT 2003-04 p9 Q25</mdoxml> 2 3 0 0 0 1 0 After the first two terms in a sequence, every term is the sum of the two previous ones. If the fifth term is 2004, what is the maximum possible value of the first term? Secondary Mapping Document DisplayCabinet Secondary Mapping Document Patterns and sequences US