Competition problem — Algebra — Maths Olympiad Prep

Library / /111 of 168

Algebra Difficulty 2.1 Junior Find the answer

Alicia starts a sequence with $m=3$ . What is the fifth term of her sequence following the algorithm: Step 1: Alicia writes down the number $m$ as the first term. Step 2: If $m$ is even, Alicia sets $n= rac{1}{2} m$ . If $m$ is odd, Alicia sets $n=m+1$ . Step 3: Alicia writes down the number $m+n+1$ as the next term. Step 4: Alicia sets $m$ equal to the value of the term that she just wrote down in Step 3. Step 5: Alicia repeats Steps 2, 3, 4 until she has five terms, at which point she stops.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

We follow Alicia's algorithm carefully: Step 1: Alicia writes down $m=3$ as the first term. Step 2: Since $m=3$ is odd, Alicia sets $n=m+1=4$ . Step 3: Alicia writes down $m+n+1=8$ as the second term. Step 4: Alicia sets $m=8$ . Step 2: Since $m=8$ is even, Alicia sets $n= rac{1}{2} m=4$ . Step 3: Alicia writes down $m+n+1=13$ as the third term. Step 4: Alicia sets $m=13$ . Step 2: Since $m=13$ is odd, Alicia sets $n=m+1=14$ . Step 3: Alicia writes down $m+n+1=28$ as the fourth term. Step 4: Alicia sets $m=28$ . Step 2: Since $m=28$ is even, Alicia sets $n= rac{1}{2} m=14$ . Step 3: Alicia writes down $m+n+1=43$ as the fifth term. Step 5: Since Alicia has written down five terms, she stops. Therefore, the fifth term is 43.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.