Math Problem Statement
What number completes the pattern? The given grid of numbers shows three rows, each following a pattern of basic arithmetic operations (addition, subtraction, multiplication, or division). The last number in the third row is missing.
Solution
The image presents a 3x4 grid of numbers with the following pattern:
9 & 1 & 6 & 4 \\ 5 & 8 & 7 & 2 \\ 1 & 3 & 5 & ? \\ \end{matrix}$$ We need to identify the missing number, denoted by `?`, based on an algebraic equation that holds across all rows, using basic arithmetic operations (addition, subtraction, multiplication, and division). ### Let's analyze each row: 1. **First row:** $$9, 1, 6, 4$$ $$9 - 1 = 8, \quad 8 - 6 = 2, \quad 2 + 4 = 6$$ There doesn't seem to be a consistent pattern in this approach. 2. **Second row:** $$5, 8, 7, 2$$ $$5 + 8 = 13, \quad 13 - 7 = 6, \quad 6 - 2 = 4$$ 3. **Third row (incomplete):** $$1, 3, 5, ?$$ We can attempt to apply the same logic or use a different arithmetic pattern to complete the third row. Let me attempt different operations and see if we can find a common relationship. I'll compute this step.For the known rows: 1. **First row:** $$9 + 1 + 6 + 4 = 20, \quad 9 \times 1 \times 6 \times 4 = 216$$ 2. **Second row:** $$5 + 8 + 7 + 2 = 22, \quad 5 \times 8 \times 7 \times 2 = 560$$ There seems to be no direct multiplication or addition relationship that consistently applies across both rows. However, a possible trend in the sum of each row suggests that the missing number in the third row might follow a similar logic. For the third row: $$1 + 3 + 5 + ?$$, we need to match a sum close to 22 or another arithmetic relationship. Let me compute the missing value based on this pattern.The missing number in the third row should be **13** in order to maintain the pattern of sums close to 22 across the rows. Thus, the complete third row would be: $$1, 3, 5, 13$$ Would you like more details on how this was calculated or have any other questions? Here are 5 related questions to explore further: 1. Could this pattern be extended to larger grids? 2. How would the pattern change if we introduced multiplication or division in between the elements? 3. What if the rows followed a geometric rather than arithmetic progression? 4. Can we derive a formula that generalizes this pattern for any 3x4 grid? 5. How do these patterns relate to number theory or sequences? **Tip:** Always look for multiple arithmetic operations (like addition and multiplication) when analyzing number patterns; a combination often reveals hidden relationships.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Arithmetic Operations
Pattern Recognition
Algebra
Formulas
Addition, subtraction, multiplication, and division patterns
Theorems
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Suitable Grade Level
Grades 6-8