If p,q are roots of the quadratic equation y2−y=4 If Sn=pn−qn,n ∈N then S15S16−S14S16−S152+S14S15S13S14

Given equation is y2−y−4=0 ..........(i)Sn=pn−qn(given) Now Sn−Sn−1 =(pn−qn) −(pn−1−qn−1) =pn−qn−pn−1+qn−1 =(pn−pn−1)−(qn−qn−1) =pn−2 (p2−p)−qn−2(q2−q).............(2)Since p,q are roots of (1) then p2−p=4, q2−q=4 Using (2) Sn−Sn−1=pn−2 (4) −qn−2(4) ⇒Sn−Sn−1=4(pn−2−qn−2)..........(3)∴Sn−Sn−1=4Sn−2To findout S15,S16−S14.S16−S152+S14S15S13.S14=S16(S15−S14)−S15(S15−S14)S13−S14=(S15−S14)(S16−S15)S13S14 Using (3) 4(S13)4(S14)S13S14=4(4)=16

If p,q are roots of the quadratic equation y2−y=4 If Sn=pn−qn,n ∈N then S15S16−S14S16−S152+S14S15S13S14

Given equation is y2−y−4=0 ..........(i)Sn=pn−qn(given) Now Sn−Sn−1 =(pn−qn) −(pn−1−qn−1) =pn−qn−pn−1+qn−1 =(pn−pn−1)−(qn−qn−1) =pn−2 (p2−p)−qn−2(q2−q).............(2)Since p,q are roots of (1) then p2−p=4, q2−q=4 Using (2) Sn−Sn−1=pn−2 (4) −qn−2(4) ⇒Sn−Sn−1=4(pn−2−qn−2)..........(3)∴Sn−Sn−1=4Sn−2To findout S15,S16−S14.S16−S152+S14S15S13.S14=S16(S15−S14)−S15(S15−S14)S13−S14=(S15−S14)(S16−S15)S13S14 Using (3) 4(S13)4(S14)S13S14=4(4)=16

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If p,q are roots of the quadratic equation $y^{2} - y = 4$ If $S_{n} = p^{n} - q^{n}, n \in N$ then $\frac{S_{15} S_{16} - S_{14} S_{16} - S_{15}^{2} + S_{14} S_{15}}{S_{13} S_{14}}$

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Detailed Solution

Given equation is $y^{2} - y - 4 = 0 .......... (i)$

$S_{n} = p^{n} - q^{n} (g i v e n) N o w S_{n} - S_{n - 1} = (p^{n} - q^{n}) - (p^{n - 1} - q^{n - 1}) = p^{n} - q^{n} - p^{n - 1} + q^{n - 1} = (p^{n} - p^{n - 1}) - (q^{n} - q^{n - 1}) = p^{n - 2} (p^{2} - p) - q^{n - 2} (q^{2} - q) ............. (2)$

Since $p, q$ are roots of (1) then $p^{2} - p = 4, q^{2} - q = 4$
Using (2) $S_{n} - S_{n - 1} = p^{n - 2} (4) - q^{n - 2} (4)$

$\begin{array}{l} \Rightarrow S_{n} - S_{n - 1} = 4 (p^{n - 2} - q^{n - 2}) .......... (3) \\ ∴ S_{n} - S_{n - 1} = 4 S_{n - 2} \\ T o f i n d o u t \frac{S_{15}, S_{16} - S_{14} . S_{16} - S_{15}^{2} + S_{14} S_{15}}{S_{13} . S_{14}} \end{array}$

$= \frac{S_{16} (S_{15} - S_{14}) - S_{15} (S_{15} - S_{14})}{S_{13} - S_{14}} = \frac{(S_{15} - S_{14}) (S_{16} - S_{15})}{S_{13} S_{14}}$
Using (3) $\frac{4 (S_{13}) 4 (S_{14})}{S_{13} S_{14}} = 4 (4) = 16$