Multiple choice

R1 and R2 are the roots of the quadratic equation x2 - x - 1 = 0. What is the value of R18 + R28?

  1. 47

  2. 1

  3. -1

  4. -47
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A Correct answer
Explanation

For x^2 - x - 1 = 0, R1 + R2 = 1 and R1 * R2 = -1. Let S_n = R1^n + R2^n. Using the recurrence S_n = S_{n-1} + S_{n-2}: S_1 = 1, S_2 = (R1+R2)^2 - 2R1R2 = 1 - 2(-1) = 3. S_3 = 3 + 1 = 4, S_4 = 4 + 3 = 7, S_5 = 11, S_6 = 18, S_7 = 29, S_8 = 47.

AI explanation

Since R1 and R2 are the roots of x^2 - x - 1 = 0, any root R satisfies the recurrence relation R^n = R^(n-1) + R^(n-2). We first calculate the initial sequence values as R1^1 = 1, R2^1 = 1, R1^2 + R2^2 = 3, and R1^3 + R2^3 = 4. Continuing this recurrence to the eighth power, the sums are R1^4 + R2^4 = 7, R1^5 + R2^5 = 11, R1^6 + R2^6 = 18, R1^7 + R2^7 = 29, and finally R1^8 + R2^8 = 47. Therefore, the value of R1^8 + R2^8 is 47.

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