Insert three numbers between $1$ and $256$ so that the resulting sequence is a $G.P.$
Insert three numbers between $1$ and $256$ so that the resulting sequence is a $G.P.$
Let $G_{1}, G_{2}, G_{3}$ be three numbers between $1$ and $256$ such that $1, G _{1}, G _{2}, G _{3}, 256$ is a $G.P.$
Therefore $\quad 256=r^{4}$ giving $r=\pm 4$ (Taking real roots only)
For $r=4,$ we have $G _{1}=a r=4, G _{2}=a r^{2}=16, G _{3}=a r^{3}=64$
Similarly, for $r=-4,$ numbers are $-4,16$ and $-64$ Hence, we can insert $4,16,64$ between $1$ and $256$ so that the resulting sequences are in $G.P.$
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- [JEE MAIN 2022]