MathLabs

Problem 5

Find all real numbers aa for which there exist non-negative real numbers x1,x2,x3,x4,x5x_1,x_2,x_3,x_4,x_5 satisfying ∑k=15kxk=a\sum_{k=1}^5kx_k=a, ∑k=15k3xk=a2\sum_{k=1}^5k^3x_k=a^2, and ∑k=15k5xk=a3\sum_{k=1}^5k^5x_k=a^3.
Step 1 of 4: Combine the three equations
Previous stepNext step
a2∑k=15kxk−2a∑k=15k3xk+∑k=15k5xk=0a^2\sum_{k=1}^5kx_k-2a\sum_{k=1}^5k^3x_k+\sum_{k=1}^5k^5x_k=0
Detailed analysis

Multiply the first relation by a^2, the second by -2a, and add the third. The right-hand side is zero. Collecting the coefficient of each x_k produces the sum shown in the next step.