First slide

Arithmetic progression

Question

$If x_{1}, x_{2}, \dots x_{n} and \frac{1}{h_{1}}, \frac{1}{h_{2}}, . \frac{1}{h_{n}} are two A.P's such that x_{3} = h_{2} = 8 and x_{8} = h_{7} = 20$ $then x_{5} h_{10} is$

Difficult

A

2560
B

2650
C

3200
D

1600

Solution

$d_{1} is common difference of A.P of x_{1}, x_{2} \dots x_{n} then$
$\begin{aligned} x_{8} - x_{3} = 5 d_{1} = 12 \Rightarrow d_{1} = 2.4 \Rightarrow x_{5} = x_{3} + 2 d_{1} = 8 + 2 . (\frac{12}{5}) = 12.8 \\ Suppose d_{2} is the common difference of an A.P \frac{1}{h_{1}}, \frac{1}{h_{2}}, \cdot \cdot \frac{1}{h_{n}} \end{aligned}$
$\begin{array}{l} \frac{1}{h_{7}} - \frac{1}{h_{2}} = 5 d_{2} = \frac{1}{20} - \frac{1}{8} = - 3 / 40 \\ d_{2} = \frac{- 3}{200} \\ ∵ \frac{1}{h_{10}} = \frac{1}{h_{7}} + 3 d_{2} = \frac{1}{20} - \frac{9}{200} = \frac{1}{200} \Rightarrow h_{10} = 200 \\ ∴ x_{5} \cdot h_{10} = 12.8 \times 200 = 2560 \end{array}$

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