Let xₙ, yₙ, zₙ, wₙ denotes n^{th} terms of four different arithmatic progressions with

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8. Sequences and Series

normal

Let $x_n, y_n, z_n, w_n$ denotes $n^{th}$ terms of four different arithmatic progressions with positive terms. If $x_4 + y_4 + z_4 + w_4 = 8$ and $x_{10} + y_{10} + z_{10} + w_{10} = 20,$ then maximum value of $x_{20}.y_{20}.z_{20}.w_{20}$ is-

$10^4$

$10^6$

$10^8$

$10^{10}$

Solution

$\because \operatorname{sum}$ of two $\mathrm{A} . \mathrm{P}^{\prime}$'s is also an $\mathrm{A} . \mathrm{P}.$

$\therefore {{\rm{x}}_4} + {{\rm{y}}_4} + {{\rm{z}}_4} + {{\rm{w}}_4} = {\rm{A}} + 3{\rm{D}} = 8$ and

$\mathrm{x}_{10}+\mathrm{y}_{10}+\mathrm{z}_{10}+\mathrm{w}_{10}=\mathrm{A}+9 \mathrm{D}=20$

$\Rightarrow A=2, D=2$

$\Rightarrow \mathrm{x}_{20}+\mathrm{y}_{20}+\mathrm{z}_{20}+\mathrm{w}_{20}=\mathrm{A}+19 \mathrm{D}=40$

Now $\frac{\mathrm{x}_{20}+\mathrm{y}_{20}+\mathrm{z}_{20}+\mathrm{w}_{20}}{4} \geq\left(\mathrm{x}_{20} \cdot \mathrm{y}_{20} \cdot \mathrm{z}_{20} \cdot \mathrm{w}_{20}\right)^{1 / 4}$

$ \Rightarrow \left( {{{\rm{x}}_{20}} \cdot {{\rm{y}}_{20}} \cdot {{\rm{z}}_{20}} \cdot {{\rm{w}}_{20}}} \right) \le {10^4}$

Standard 11

Mathematics

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Let $x_n, y_n, z_n, w_n$ denotes $n^{th}$ terms of four different arithmatic progressions with positive terms. If $x_4 + y_4 + z_4 + w_4 = 8$ and $x_{10} + y_{10} + z_{10} + w_{10} = 20,$ then maximum value of $x_{20}.y_{20}.z_{20}.w_{20}$ is-

Solution

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