Math riddles with visual solution
5 challenging math problem
Prompt
solve this with solution, also visual this problem and solving best as you can, create a page and use math format to show question and answer in proper form: a + b + c = 2 , a^2 + b^2 + c^2 = 10 , a^3 + b^3 + c^3 = 23 , a^5 + b^5 + c^5 = ?
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Answer guidance
Answer: 207. solution: Method: Newton's Sums To solve this problem, we will use Newton's Sums (also known as Newton's identities). These identities provide a relationship between two types of symmetric polynomials: power sums and elementary symmetric polynomials. Let's define our terms: 1. Power Sums (p_k) p_k = a^k + b^k + c^k From the problem statement, we have: p_1 = a + b + c = 2 p_2 = a^2 + b^2 + c^2 = 10 p_3 = a^3 + b^3 + c^3 = 23 2. Elementary Symmetric Polynomials (e_k) e_1 = a + b + c e_2 = ab + ac + bc e_3 = abc Step-by-Step Calculation Our strategy is to first find the values of the elementary symmetric polynomials (e_1, e_2, e_3) and then use them to calculate the higher power sums (p_4 and finally p_5). Step 1: Find the value of e_1 By definition, e_1 is the same as p_1. e_1 = a + b + c = 2 Step 2: Find the value of e_2 We use the identity that relates p_1, p_2, and e_2: p_1^2 = (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) p_1^2 = p_2 + 2 * e_2 Now, we can solve for e_2: 2 * e_2 = p_1^2 - p_2 2 * e_2 = (2)^2 - 10 2 * e_2 = 4 - 10 2 * e_2 = -6 e_2 = -3 Step 3: Find the value of e_3 We use the third Newton's identity: p_3 - e_1 * p_2 + e_2 * p_1 - 3 * e_3 = 0 Substitute the known values to solve for e_3: 23 - (2)(10) + (-3)(2) - 3 * e_3 = 0 23 - 20 - 6 - 3 * e_3 = 0 -3 - 3 * e_3 = 0 -3 = 3 * e_3 e_3 = -1 Step 4: Find the value of p_4 = a^4 + b^4 + c^4 Now that we have all the elementary symmetric polynomials, we can find the higher power sums. We use the fourth Newton's identity: p_4 - e_1 * p_3 + e_2 * p_2 - e_3 * p_1 = 0 Solve for p_4: p_4 = e_1 * p_3 - e_2 * p_2 + e_3 * p_1 p_4 = (2)(23) - (-3)(10) + (-1)(2) p_4 = 46 + 30 - 2 p_4 = 74 Step 5: Find the value of p_5 = a^5 + b^5 + c^5 Finally, we use the fifth Newton's identity to find our target value, p_5: p_5 - e_1 * p_4 + e_2 * p_3 - e_3 * p_2 = 0 Solve for p_5: p_5 = e_1 * p_4 - e_2 * p_3 + e_3 * p_2 p_5 = (2)(74) - (-3)(23) + (-1)(10) p_5 = 148 + 69 - 10 p_5 = 217 - 10 p_5 = 207 Conclusion The value of the expression is: a^5 + b^5 + c^5 = 207
