jee-main 2023 Q64

jee-main · India · session2_15apr_shift1 Geometric Sequences and Series Arithmetic-Geometric Sequence Interplay

Let $A _ { 1 }$ and $A _ { 2 }$ be two arithmetic means and $G _ { 1 } , G _ { 2 }$ and $G _ { 3 }$ be three geometric means of two distinct positive numbers. Then $G _ { 1 } ^ { 4 } + G _ { 2 } ^ { 4 } + G _ { 3 } ^ { 4 } + G _ { 1 } ^ { 2 } G _ { 3 } ^ { 2 }$ is equal to
(1) $\left( A _ { 1 } + A _ { 2 } \right) ^ { 2 } G _ { 1 } G _ { 3 }$
(2) $2 \left( A _ { 1 } + A _ { 2 } \right) G _ { 1 } G _ { 3 }$
(3) $\left( A _ { 1 } + A _ { 2 } \right) G _ { 1 } ^ { 2 } G _ { 3 } ^ { 2 }$
(4) $2 \left( A _ { 1 } + A _ { 2 } \right) G _ { 1 } ^ { 2 } G _ { 3 } ^ { 2 }$

Let $A _ { 1 }$ and $A _ { 2 }$ be two arithmetic means and $G _ { 1 } , G _ { 2 }$ and $G _ { 3 }$ be three geometric means of two distinct positive numbers. Then $G _ { 1 } ^ { 4 } + G _ { 2 } ^ { 4 } + G _ { 3 } ^ { 4 } + G _ { 1 } ^ { 2 } G _ { 3 } ^ { 2 }$ is equal to\\
(1) $\left( A _ { 1 } + A _ { 2 } \right) ^ { 2 } G _ { 1 } G _ { 3 }$\\
(2) $2 \left( A _ { 1 } + A _ { 2 } \right) G _ { 1 } G _ { 3 }$\\
(3) $\left( A _ { 1 } + A _ { 2 } \right) G _ { 1 } ^ { 2 } G _ { 3 } ^ { 2 }$\\
(4) $2 \left( A _ { 1 } + A _ { 2 } \right) G _ { 1 } ^ { 2 } G _ { 3 } ^ { 2 }$

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