Paragraph: Let $p, q$ be integers and let $\alpha, \beta$ be the roots of the equation, $x^{2}-x-1=0$, where $\alpha
eq \beta$. For $n=0,1,2, \ldots$, let $a_{n}=p \alpha^{n}+q \beta^{n}$ FACT: If $a$ and $b$ are rational numbers and $a+b \sqrt{5}=0$, then $a=0=b$. Question: If $a_{4}=28$, then…

Question

Paragraph: Let $p, q$ be integers and let $\alpha, \beta$ be the roots of the equation, $x^{2}-x-1=0$, where $\alpha 
eq \beta$. For $n=0,1,2, \ldots$, let $a_{n}=p \alpha^{n}+q \beta^{n}$ FACT: If $a$ and $b$ are rational numbers and $a+b \sqrt{5}=0$, then $a=0=b$. Question: If $a_{4}=28$, then $p+2 q=$

Aicharya · Accepted Answer

&#945; 2 = &#945; + 1 &#8658; &#945; 4 = ( &#945; 2 ) 2 &#8658; &#945; 4 = ( &#945; + 1 ) 2 &#8658; &#945; 4 = &#945; 2 + 2 &#945; + 1 &#8658; &#945; 4 = &#945; + 1 + 2 &#945; + 1 &#8658; &#945; 4 = 3 &#945; + 2 A l s o&#160;&#160; &#945; + &#946; = 1 &#8756; &#160; &#160; &#160; &#160; a 4 = 28 &#160; &#8658; &#160; &#160; p &#945; 4 + q &#946; 4 = p 3 &#945; + 2 + q 3 &#946; + 2 = 2 8 &#8658; &#160; &#160; &#160; p 3 &#945; + 2 + q 3 - 3 &#945; + 2 = 2 8 &#8658; &#160; &#160; &#160; &#945; 3 p - 3 q + 2 p + 5 q = 28 (as &#945; &#8712; Q c ) (as it is root of quadratic equation) &#8658; &#160; &#160; &#160; &#160; p = q , &#160; 2 p + 5 q = 28 &#160; &#8658; &#160; &#160; p = q = 4 &#8756; &#160; &#160; &#160; &#160; p + 2 q = 1 2

Paragraph: Let $p, q$ be integers and let $\alpha, \beta$ be the roots of the equation, $x^{2}-x-1=0$, where…

Solution