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यदि $x_{1}, x_{2}, \ldots ., x_{n}$ तथा $\frac{1}{h_{1}}, \frac{1}{h_{2}}, \ldots ., \frac{1}{h_{n}}$ दो ऐसी समांतर श्रेढियां हैं कि $x_{3}=h_{2}=8$ तथा $x_{8}=h_{7}=20$ है, तो $x_{5} . h_{10}$ का मान है
$2560$
$2650$
$3200$
$1600$
Solution
Suppose ${d_1}$ is the common difference of the $A.P.$
${x_1},{x_2},….{x_n}$ then
$\because $ ${x_8} – {x_3} = 5{d_1} = 12 \Rightarrow {d_1} = \frac{{12}}{5} = 2.4$
$ \Rightarrow {x_5} = {x_3} + 2{d_1} = 8 + 2 \times \frac{{12}}{5} = 12.8$
Suppose ${d_2}$ is the common difference of the $A.P.$ $\frac{1}{{{h_1}}},\frac{1}{{{h_2}}},…..\frac{1}{{{h_n}}}$ then
$5{d_2} = \frac{1}{{20}} – \frac{1}{8} = \frac{{ – 3}}{{40}} \Rightarrow {d_2} = \frac{{ – 3}}{{200}}$
$\because$ $\frac{1}{{{h_{10}}}} = \frac{1}{{{h_7}}} + 3{d_2} = \frac{1}{{200}} \Rightarrow {h_{10}} = 200$
$ \Rightarrow {x_5}.{h_{10}} = 12.8 \times 200 = 2560$
