Number theoryDifficulty 2.7Find the answerCEMC Pascal · Canada · 2023
Starting with a positive integer m, Alicia creates a sequence by applying the following algorithm:
Step 1: Alicia writes down the number m as the first term of the sequence. Step 2: If m is even, Alicia sets n=21m. If m is odd, Alicia sets n=m+1. Step 3: Alicia writes down the number m+n+1 as the next term of the sequence. 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.
For example, starting with m=1, Alicia’s sequence would be 1, 4, 7, 16, 25.
Alicia starts a sequence with m=3. What is the fifth term of her sequence?
A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.
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 $n = 21m = 4$. 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 $n = 21m = 14$. Step 3: Alicia writes down $m + n + 1 = 43$ as the fifth term. Step 5: Since Alicia has written down five terms, she stops.