JEE Main 2020 (Online) 7th January Evening ShiftQuestion 1
Let $\alpha$ and $\beta$ be the roots of the equation $x^2-x-1=0$. If $p_k=(\alpha)^k+(\beta)^k$, $k\ge1$, then which one of the following statements is not true?
Quadratic Equation · Powers of Roots (αⁿ ± βⁿ Recurrence)
Solution
Concept. $x^2=x+1$ gives $p_k=p_{k-1}+p_{k-2}$; list the first few values. Formulas used. (1) $p_k=p_{k-1}+p_{k-2}$, $p_1=1$, $p_2=(\alpha+\beta)^2-2\alpha\beta=3$Step 1 - Values. By (1): $p_3=4$, $p_4=7$, $p_5=11$. Step 2 - Check each. (A) $p_5-p_4=4=p_3$: true. (B) $p_5=11$: true. (C) $1+3+4+7+11=26$: true. (D) $p_2p_3=12\ne11$: not true. Answer: (D)