Multiple choice

If $\alpha, \beta$ are the roots of the equation $x^2-ax+b=0$ and $A_n=\alpha ^n+\beta ^n$, then which of the following is true?

  1. $A_{n+1}=aA_n+bA_{n-1}$
  2. $A_{n+1}=bA_n+aA_{n-1}$
  3. $A_{n+1}=aA_n-bA_{n-1}$
  4. $A_{n+1}=bA_n-aA_{n-1}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Since alpha and beta are roots of x^2 - ax + b = 0, we have alpha^2 = a*alpha - b and beta^2 = a*beta - b. Multiplying by alpha^(n-1) and beta^(n-1) respectively and adding gives alpha^(n+1) + beta^(n+1) = a(alpha^n + beta^n) - b(alpha^(n-1) + beta^(n-1)), which is A(n+1) = a*A(n) - b*A(n-1).

AI explanation

Since alpha and beta are roots of x squared minus a x plus b equals 0, they satisfy alpha squared equals a alpha minus b and beta squared equals a beta minus b. For the sequence A sub n equals alpha to the power of n plus beta to the power of n, multiply the equation for A sub n by alpha or beta to express higher powers. We have alpha to the power of n plus 1 equals a alpha to the power of n minus b alpha to the power of n minus 1, and similarly for beta. Adding these two equations gives A sub n plus 1 equals a times A sub n minus b times A sub n minus 1.

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