Let are the roots of the equation, x 2 - x - 1 = 0 and S n = 2023 n + 2024 n , then : [2024]

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Q.
Let $α, β$ are the roots of the equation, $x^{2} - x - 1 = 0$ and $S_{n} = 2023 α^{n} + 2024 β^{n},$ then : [2024]

1 $2 S_{12} = S_{11} + S_{10}$

2 $2 S_{11} = S_{12} + S_{10}$

3 $S_{11} = S_{10} + S_{12}$

4 $S_{12} = S_{11} + S_{10}$

Ans.
(4)

Given, $α$ and $β$ are the roots of $x^{2} - x - 1 = 0$

$\Rightarrow α^{2} - α - 1 = 0$ and $β^{2} - β - 1 = 0$

$\Rightarrow α^{2} = 1 + α$ and $β^{2} = 1 + β$ (i)

Now, $S_{10} = 2023 α^{10} + 2024 β^{10} [∵ S_{n} = 2023 α^{n} + 2024 β^{n}]$

$S_{11} = 2023 α^{11} + 2024 β^{11}$

$\Rightarrow S_{10} + S_{11} = 2023 α^{10} (1 + α) + 2024 β^{10} (1 + β)$

$= 2023 α^{10} \cdot α^{2} + 2024 β^{10} \cdot β^{2} [From (i)]$

$= 2023 α^{12} + 2024 β^{12} = S_{12}$

$∴ S_{12} = S_{10} + S_{11}$

Solution

Q.
Let $α, β$ are the roots of the equation, $x^{2} - x - 1 = 0$ and $S_{n} = 2023 α^{n} + 2024 β^{n},$ then : [2024]

1 $2 S_{12} = S_{11} + S_{10}$

2 $2 S_{11} = S_{12} + S_{10}$

3 $S_{11} = S_{10} + S_{12}$

4 $S_{12} = S_{11} + S_{10}$

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Solution

Q. Let α,β are the roots of the equation, x2-x-1=0 and Sn=2023αn+2024βn, then : [2024]

1 2S12=S11+S10

2 2S11=S12+S10

3 S11=S10+S12

4 S12=S11+S10

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Q.
Let $α, β$ are the roots of the equation, $x^{2} - x - 1 = 0$ and $S_{n} = 2023 α^{n} + 2024 β^{n},$ then : [2024]

1 $2 S_{12} = S_{11} + S_{10}$

2 $2 S_{11} = S_{12} + S_{10}$

3 $S_{11} = S_{10} + S_{12}$

4 $S_{12} = S_{11} + S_{10}$