Multiple choice

If $\alpha , \beta $ are roots of equation $x^{2}-4x-3=0$ and $s_{n}= \alpha ^{n} + \beta ^{n},n \in N$, then the value of $\displaystyle \frac{s_{7}-4s_{6}}{s_{5}}$ is

  1. $3$
  2. $4$
  3. $5$
  4. $7$
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A Correct answer
Explanation

Since alpha and beta are roots of x^2 - 4x - 3 = 0, alpha^2 = 4*alpha + 3 and beta^2 = 4*beta + 3. Multiplying by alpha^(n-2) and beta^(n-2) and adding gives s_n = 4*s_(n-1) + 3*s_(n-2). Thus, s_7 - 4*s_6 = 3*s_5. The ratio is 3.

AI explanation

Because alpha and beta are the roots of x^2 - 4x - 3 = 0, each root satisfies the equation, giving the recurrence relation x^n = 4x^(n-1) + 3x^(n-2). Multiplying by x^(5-n) gives x^7 = 4x^6 + 3x^5, which applies to both alpha and beta. Adding these two equations yields the relation s_7 = 4s_6 + 3s_5. Rearranging this result gives (s_7 - 4s_6) / s_5 = 3.

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