Multiple choice

Let $\alpha$ and $\beta$ be the roots of the equation $x^2 - x - 1 = 0$. If $p_k = (\alpha)^k + (\beta)^k , k \ge 1$, then which one of the following statements is not true ?

  1. $p_3 = p_5 - p_4$
  2. $p_5 = p_2 . p_3$
  3. $p_5 = 11$
  4. $(p_1 + p_2 + p_3 + p_4 + p_5) = 26$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The roots of the equation satisfy the relation p_{k+2} = p_{k+1} + p_k. Calculating the values gives p_1 = 1, p_2 = 3, p_3 = 4, p_4 = 7, and p_5 = 11. Statement B claims p_5 = p_2 * p_3, which is 11 = 12, and is therefore false.

AI explanation

From the equation x^2 - x - 1 = 0, the sum of the roots is a + b = 1 and the product is ab = -1. We calculate the sequence values: p1 = 1, p2 = 3, p3 = 4, p4 = 7, and p5 = 11. Evaluating the statements shows p5 equals 11, the sum p1 + p2 + p3 + p4 + p5 equals 26, and p3 equals p5 - p4. However, p2 * p3 evaluates to 12, which does not equal p5, making the statement p5 = p2 * p3 false.

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