jee-main 2024 Q61

jee-main · India · session1_27jan_shift2 Sequences and series, recurrence and convergence Multiple-choice on sequence properties
If $\alpha , \beta$ are the roots of the equation, $\mathrm { x } ^ { 2 } - \mathrm { x } - 1 = 0$ and $\mathrm { S } _ { \mathrm { n } } = 2023 \alpha ^ { \mathrm { n } } + 2024 \beta ^ { \mathrm { n } }$, then
(1) $2 \quad \mathrm {~S} _ { 12 } = \mathrm { S } _ { 11 } + \mathrm { S } _ { 10 }$
(2) $\mathrm { S } _ { 12 } = \mathrm { S } _ { 11 } + \mathrm { S } _ { 10 }$
(3) $2 \mathrm {~S} _ { 11 } = \mathrm { S } _ { 12 } + \mathrm { S } _ { 10 }$
(4) $\mathrm { S } _ { 11 } = \mathrm { S } _ { 10 } + \mathrm { S } _ { 12 }$ 
If $\alpha , \beta$ are the roots of the equation, $\mathrm { x } ^ { 2 } - \mathrm { x } - 1 = 0$ and $\mathrm { S } _ { \mathrm { n } } = 2023 \alpha ^ { \mathrm { n } } + 2024 \beta ^ { \mathrm { n } }$, then\\
(1) $2 \quad \mathrm {~S} _ { 12 } = \mathrm { S } _ { 11 } + \mathrm { S } _ { 10 }$\\
(2) $\mathrm { S } _ { 12 } = \mathrm { S } _ { 11 } + \mathrm { S } _ { 10 }$\\
(3) $2 \mathrm {~S} _ { 11 } = \mathrm { S } _ { 12 } + \mathrm { S } _ { 10 }$\\
(4) $\mathrm { S } _ { 11 } = \mathrm { S } _ { 10 } + \mathrm { S } _ { 12 }$
Paper Questions
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