Multiple choice

If $\alpha $ and $\beta $ are the roots of equation ${x^2} - 3x + 1 = 0$ and ${a_n} = {\alpha ^n} + {\beta ^n},n \in N$ then the value of $\dfrac{{{a_7} + {a_5}}}{{{a_6}}}$

  1. $1$
  2. $2$
  3. $3$
  4. $4$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Roots of x^2 - 3x + 1 = 0 satisfy alpha^2 - 3alpha + 1 = 0, so alpha^2 + 1 = 3alpha. Multiply by alpha^(n-1): alpha^(n+1) + alpha^(n-1) = 3 * alpha^n. Similarly for beta. Adding them: a_(n+1) + a_(n-1) = 3 * a_n. For n=6, a_7 + a_5 = 3 * a_6. Thus (a_7 + a_5) / a_6 = 3.

AI explanation

Since alpha and beta are roots of x^2 - 3x + 1 = 0, they satisfy the equation x^2 = 3x - 1. Multiplying by x^(n-2) gives the recurrence relation a_n = 3a_(n-1) - a_(n-2) for the sequence a_n = alpha^n + beta^n. Adding the equations for a_7 and a_5 yields a_7 + a_5 = 3a_6 - a_5 + 3a_4 - a_3, and using the recurrence again simplifies this to 3(a_6 - a_4) + (a_4 - a_3) = 3a_6. Therefore, the ratio (a_7 + a_5) / a_6 equals 3.

Keep practicing — related questions