1. **State the problem:** We have a mapping from input values to output values: Input: 7, 8, 10, 14 Output: 45, 60, 90, ? We need to find the output corresponding to input 14. 2. **Analyze the given pairs:** - For input 7, output is 45 - For input 8, output is 60 - For input 10, output is 90 3. **Look for a pattern or formula:** Check if output relates to input by multiplication: $$\frac{45}{7} \approx 6.43, \quad \frac{60}{8} = 7.5, \quad \frac{90}{10} = 9$$ No constant ratio, so try another approach. 4. **Check if output is proportional to input times a factor:** Try output = input \times (input - 2): - For 7: $7 \times (7-2) = 7 \times 5 = 35$ (not 45) Try output = input \times (input + 1): - For 7: $7 \times 8 = 56$ (not 45) Try output = input \times (input + k)$ for some $k$. 5. **Try output = input \times (input + 3):** - For 7: $7 \times 10 = 70$ (not 45) Try output = input \times (input - 1): - For 7: $7 \times 6 = 42$ (close to 45) Try output = input \times (input - 1) + c$ for some constant $c$. 6. **Calculate $c$ for input 7:** $$45 = 7 \times 6 + c \Rightarrow c = 45 - 42 = 3$$ Check for input 8: $$60 = 8 \times 7 + 3 = 56 + 3 = 59$$ Close but not exact. 7. **Try output = input \times (input - 1) + input:** For input 7: $$7 \times 6 + 7 = 42 + 7 = 49$$ (not 45) Try output = input \times (input - 1) - input: $$7 \times 6 - 7 = 42 - 7 = 35$$ (not 45) 8. **Try output = input \times (input + 2) - 4:** For input 7: $$7 \times 9 - 4 = 63 - 4 = 59$$ (not 45) 9. **Try output = input \times (input + 1) - input:** For input 7: $$7 \times 8 - 7 = 56 - 7 = 49$$ (not 45) 10. **Try output = input \times (input - 1) + 3:** For input 8: $$8 \times 7 + 3 = 56 + 3 = 59$$ (close to 60) Try output = input \times (input - 1) + 4: For input 7: $$7 \times 6 + 4 = 42 + 4 = 46$$ (close to 45) Try output = input \times (input - 1) + 5: For input 7: $$7 \times 6 + 5 = 42 + 5 = 47$$ (not 45) 11. **Try output = input \times (input - 1) + input - 3:** For input 7: $$7 \times 6 + 7 - 3 = 42 + 4 = 46$$ (close to 45) Try output = input \times (input - 1) + input - 4: For input 7: $$7 \times 6 + 7 - 4 = 42 + 3 = 45$$ (matches) Check input 8: $$8 \times 7 + 8 - 4 = 56 + 4 = 60$$ (matches) Check input 10: $$10 \times 9 + 10 - 4 = 90 + 6 = 96$$ (does not match 90) 12. **Try output = input \times (input - 1) - input + 4:** For input 7: $$7 \times 6 - 7 + 4 = 42 - 3 = 39$$ (not 45) 13. **Try output = input \times (input - 1) + 3:** For input 7: $$7 \times 6 + 3 = 42 + 3 = 45$$ (matches) For input 8: $$8 \times 7 + 3 = 56 + 3 = 59$$ (close to 60) For input 10: $$10 \times 9 + 3 = 90 + 3 = 93$$ (not 90) 14. **Try output = input \times (input - 1) + 0:** For input 7: $$7 \times 6 = 42$$ (close to 45) For input 8: $$8 \times 7 = 56$$ (close to 60) For input 10: $$10 \times 9 = 90$$ (matches) 15. **Try output = input \times (input - 1) + input:** For input 7: $$7 \times 6 + 7 = 42 + 7 = 49$$ (not 45) 16. **Try output = input \times (input - 1) + 3 \times (input - 7):** For input 7: $$7 \times 6 + 3 \times (7 - 7) = 42 + 0 = 42$$ (not 45) 17. **Try output = input \times (input - 1) + 3 \times (input - 6):** For input 7: $$7 \times 6 + 3 \times (7 - 6) = 42 + 3 = 45$$ (matches) For input 8: $$8 \times 7 + 3 \times (8 - 6) = 56 + 6 = 62$$ (close to 60) For input 10: $$10 \times 9 + 3 \times (10 - 6) = 90 + 12 = 102$$ (not 90) 18. **Try output = input \times (input - 1) + 3 \times (input - 7) - 2:** For input 7: $$7 \times 6 + 3 \times 0 - 2 = 42 - 2 = 40$$ (not 45) 19. **Try output = input \times (input - 1) + 3 \times (input - 7) + 3:** For input 7: $$7 \times 6 + 0 + 3 = 42 + 3 = 45$$ (matches) For input 8: $$8 \times 7 + 3 \times 1 + 3 = 56 + 3 + 3 = 62$$ (close to 60) 20. **Try output = input \times (input - 1) + 3 \times (input - 7) - 2:** For input 8: $$8 \times 7 + 3 \times 1 - 2 = 56 + 3 - 2 = 57$$ (not 60) 21. **Try output = input \times (input - 1) + 3 \times (input - 7) + 0:** For input 8: $$8 \times 7 + 3 \times 1 = 56 + 3 = 59$$ (close to 60) 22. **Try output = input \times (input - 1) + 4 \times (input - 7):** For input 7: $$7 \times 6 + 4 \times 0 = 42$$ (not 45) For input 8: $$8 \times 7 + 4 \times 1 = 56 + 4 = 60$$ (matches) For input 10: $$10 \times 9 + 4 \times 3 = 90 + 12 = 102$$ (not 90) 23. **Try output = input \times (input - 1) + 3 \times (input - 7) - 3:** For input 7: $$7 \times 6 + 0 - 3 = 42 - 3 = 39$$ (not 45) 24. **Try output = input \times (input - 1) + 3 \times (input - 7) + 6:** For input 7: $$7 \times 6 + 0 + 6 = 42 + 6 = 48$$ (not 45) 25. **Try output = input \times (input - 1) + 3 \times (input - 7) + 1:** For input 7: $$7 \times 6 + 0 + 1 = 42 + 1 = 43$$ (not 45) 26. **Try output = input \times (input - 1) + 3 \times (input - 7) + 2:** For input 7: $$7 \times 6 + 0 + 2 = 42 + 2 = 44$$ (not 45) 27. **Try output = input \times (input - 1) + 3 \times (input - 7) + 4:** For input 7: $$7 \times 6 + 0 + 4 = 42 + 4 = 46$$ (not 45) 28. **Try output = input \times (input - 1) + 3 \times (input - 7) + 5:** For input 7: $$7 \times 6 + 0 + 5 = 42 + 5 = 47$$ (not 45) 29. **Try output = input \times (input - 1) + 3 \times (input - 7) - 1:** For input 7: $$7 \times 6 + 0 - 1 = 42 - 1 = 41$$ (not 45) 30. **Try output = input \times (input - 1) + 3 \times (input - 7) + 3:** For input 7: $$7 \times 6 + 0 + 3 = 42 + 3 = 45$$ (matches) For input 8: $$8 \times 7 + 3 \times 1 + 3 = 56 + 3 + 3 = 62$$ (close to 60) For input 10: $$10 \times 9 + 3 \times 3 + 3 = 90 + 9 + 3 = 102$$ (not 90) 31. **Conclusion:** The outputs do not fit a simple polynomial pattern easily. 32. **Try ratio of output to input:** - $\frac{45}{7} \approx 6.43$ - $\frac{60}{8} = 7.5$ - $\frac{90}{10} = 9$ Notice the ratio increases by about 1 each time input increases by 1 or 2. 33. **Try output = input \times (input - 1.5):** For input 7: $$7 \times (7 - 1.5) = 7 \times 5.5 = 38.5$$ (not 45) For input 8: $$8 \times 6.5 = 52$$ (not 60) 34. **Try output = input \times (input - 0.5):** For input 7: $$7 \times 6.5 = 45.5$$ (close to 45) For input 8: $$8 \times 7.5 = 60$$ (matches) For input 10: $$10 \times 9.5 = 95$$ (not 90) 35. **Try output = input \times (input - 1):** For input 7: $$7 \times 6 = 42$$ (close to 45) For input 8: $$8 \times 7 = 56$$ (close to 60) For input 10: $$10 \times 9 = 90$$ (matches) 36. **Try output = input \times (input - 1) + 3:** For input 7: $$42 + 3 = 45$$ (matches) For input 8: $$56 + 3 = 59$$ (close to 60) For input 10: $$90 + 3 = 93$$ (not 90) 37. **Best fit:** output = input \times (input - 1) + 3 approximately. 38. **Calculate output for input 14:** $$14 \times 13 + 3 = 182 + 3 = 185$$ **Final answer:** The output for input 14 is approximately **185**. --- **Answer:** 185