Mathematical Analysis of Power Sum Expressions

System of Power Sum Equations

• The provided transcript presents a series of equations involving three variables ($a$, $b$, and $c$) raised to successive integer powers.
• The objective is to determine the value of the fourth power sum expression, denoted as $a^4 + b^4 + c^4$, based on the given constraints.
• Fundamental Equations Provided:
  • First Power Sum ($p_1$): $a + b + c = 4$
  • Second Power Sum ($p_2$): $a^2 + b^2 + c^2 = 10$
  • Third Power Sum ($p_3$): $a^3 + b^3 + c^3 = 22$
  • Target Expression ($p_4$): $a^4 + b^4 + c^4 = ?$

Mathematical Framework: Elementary Symmetric Polynomials and Newton's Sums

• To solve this system efficiently without finding the individual values of $a$, $b$, and $c$, we employ the theory of symmetric polynomials.
• Definitions of Elementary Symmetric Polynomials ($e_k$):
  • $e_1 = a + b + c$
  • $e_2 = ab + bc + ca$
  • $e_3 = abc$
• Definitions of Power Sums ($p_k$):
  • $p_k = a^k + b^k + c^k$
• Newton-Girard Formulae:
  • These formulas relate power sums to elementary symmetric polynomials through recursive relations:
    • $p_1 - e_1 = 0$
    • $p_2 - e_1 p_1 + 2e_2 = 0$
    • $p_3 - e_1 p_2 + e_2 p_1 - 3e_3 = 0$
    • $p_4 - e_1 p_3 + e_2 p_2 - e_3 p_1 = 0$

Step-by-Step Implementation and Calculation

• Step 1: Determine the value of $e_1$

  • From the first given equation: $a + b + c = 4$.
  • Therefore, $e_1 = 4$.

• Step 2: Determine the value of $e_2$

  • We use the algebraic identity for the square of a trinomial:
    • $(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$
  • Substituting the power sum notation:
    • $e_1^2 = p_2 + 2e_2$
  • Substituting the numerical values:
    • $4^2 = 10 + 2e_2$
    • $16 = 10 + 2e_2$
    • $2e_2 = 6$
    • $e_2 = 3$

• Step 3: Determine the value of $e_3$

  • We utilize the third Newton-Girard identity:
    • $p_3 - e_1 p_2 + e_2 p_1 - 3e_3 = 0$
  • Substituting the known numerical values ($p_3 = 22$, $e_1 = 4$, $p_2 = 10$, $e_2 = 3$, $p_1 = 4$):
    • $22 - (4 \times 10) + (3 \times 4) - 3e_3 = 0$
    • $22 - 40 + 12 - 3e_3 = 0$
    • $-18 + 12 - 3e_3 = 0$
    • $-6 - 3e_3 = 0$
    • $3e_3 = -6$
    • $e_3 = -2$

• Step 4: Calculate the Target Fourth Power Sum ($p_4$)

  • We utilize the fourth Newton-Girard identity:
    • $p_4 - e_1 p_3 + e_2 p_2 - e_3 p_1 = 0$
  • Isolating $p_4$:
    • $p_4 = e_1 p_3 - e_2 p_2 + e_3 p_1$
  • Substituting all determined values ($e_1 = 4$, $p_3 = 22$, $e_2 = 3$, $p_2 = 10$, $e_3 = -2$, $p_1 = 4$):
    • $p_4 = (4 \times 22) - (3 \times 10) + (-2 \times 4)$
    • $p_4 = 88 - 30 - 8$
    • $p_4 = 58 - 8$
    • $p_4 = 50$

Final Solution Summary

• Given the initial conditions:
  • Sum of variables: $4$
  • Sum of squares: $10$
  • Sum of cubes: $22$
• The resulting sum of the variables each raised to the fourth power is:
  • $a^4 + b^4 + c^4 = 50$