Problem 24 · 2025 Math Kangaroo
Stretch
Algebra & Patterns
arithmetic-sequencesubstitution
Fritz fills out a table with two columns and 51 rows. In the first row, he writes 5 on the left and 3 on the right. In each subsequent row he writes the sum of the two numbers from the row above on the left and the positive difference of these two numbers on the right. Which two numbers does he write in the bottom row?
Show answer
Answer: D — \(5\cdot 2^{25}\) and \(3\cdot 2^{25}\)
Show hints
Hint 1 of 2
Compute a few rows and watch the left and right entries separately.
Still stuck? Show hint 2 →
Hint 2 of 2
Each pair (L, R) becomes (L+R, L−R); the left entry doubles every two rows starting from 5.
Show solution
Approach: track the recurrence two rows at a time
- Rows give left entries 5, 8, 10, 16, 20, 32, 40… and right entries 3, 2, 6, 4, 12, 8, 24…
- Odd row 2k+1 has left = 5·2k, right = 3·2k.
- Row 51 (k = 25): 5·225 and 3·225.
Mark:
· log in to save