A hard question on cycles for P4–P5, with a step-by-step solution, diagrams and the answer. Cycle questions appear in the Hua Xia Cup from the P1 first round to the P5 papers: which comes in a given place, which day of the week, which last digit.
The question
Ming writes the group 2, 1, 9 over and over in one long line:
2, 1, 9, 2, 1, 9, 2, 1, 9, …
What do the first 158 numbers add up to?
Show the hints (3)
💡Hint 1: Each group has 3 numbers. What does one group add up to?
💡Hint 2: How many whole groups are in the first 158 numbers? Which numbers are left over?
💡Hint 3: 158 ÷ 3 = 52 remainder 2: 52 groups each adding to 12, plus the left-over 2, 1.
Worked solution
Have a go first, then read the solution step by step.
1.Each group of 3 adds to 2 + 1 + 9 = 12.
2.158 ÷ 3 = 52 remainder 2: 52 groups make 52 × 12 = 624.
3.The 2 left over are the first 2 of a group: 2 + 1 = 3. In all 624 + 3 = 627.
⚠️Common mistake: adding the left-over numbers from the end of a group. They are the ones at the start.
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