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Real numbers x and y satisfy x + y = 4 and x^3 + y^3 = 28. Compute…
Question
Real numbers x and y satisfy x + y = 4 and x^3 + y^3 = 28. Compute x^5 + y^5.
✓ Verified answer: 244checked by our engine — not a guess
Step-by-step solution
Let s = x+y = 4 and q = xy. Then x^3+y^3 = s^3 - 3qs = 64 - 12q = 28, so q = 3.
Use the recurrence p_n = s·p_{n-1} - q·p_{n-2} for power sums p_n = x^n + y^n:
p1 = 4; p2 = s p1 - 2q = 16 - 6 = 10; p3 = s p2 - q p1 = 40 - 12 = 28 (checks); p4 = s p3 - q p2 = 112 - 30 = 82; p5 = s p4 - q p3 = 328 - 84 = 244.
(Indeed x,y = 1,3 give 1 + 243 = 244.)
Final answer244
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