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(D) Given the equation $x^2 - x - 1 = 0$,the roots are $\alpha$ and $\beta$. By Vieta's formulas,$\alpha + \beta = 1$ and $\alpha \beta = -1$. Since $

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$p_5 = p_2 \cdot p_3$

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(D) Given the equation $x^2 - x - 1 = 0$,the roots are $\alpha$ and $\beta$. By Vieta's formulas,$\alpha + \beta = 1$ and $\alpha \beta = -1$. Since $\alpha$ and $\beta$ are roots,they satisfy $\alpha^2 = \alpha + 1$ and $\beta^2 = \beta + 1$. Multiplying by $x^{k-2}$,we get $x^k = x^{k-1} + x^{k-2}$. Thus,$p_k = p_{k-1} + p_{k-2}$. Calculating the values: $p_1 = \alpha + \beta = 1$. $p_2 = (\alpha + \beta)^2 - 2\alpha\beta = (1)^2 - 2(-1) = 1 + 2 = 3$. $p_3 = p_2 + p_1 = 3 + 1 = 4$. $p_4 = p_3 + p_2 = 4 + 3 = 7$. $p_5 = p_4 + p_3 = 7 + 4 = 11$. Checking the options: $A: p_1 + p_2 + p_3 + p_4 + p_5 = 1 + 3 + 4 + 7 + 11 = 26$ (True). $B: p_5 = 11$ (True). $C: p_5 - p_4 = 11 - 7 = 4 = p_3$ (True). $D: p_2 \cdot p_3 = 3 \cdot 4 = 12 
eq p_5$ (False). Therefore,the statement that is not true is $p_5 = p_2 \cdot p_3$.

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

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