Question

If 10 harmonic means $H_1,H_2,H_3..........H_{10}$ are inserted between 7 and $-\frac{1}{3},$ then

• A. $H_1=-7$
• B. $H_2=\frac{3}{7}$
• C. $H_1=\frac{1}{7}$
• D. $H_{10}=-\frac{7}{19}$

Answer 1

Answer: (A), (D)

$H_{10}=-\frac{7}{19}$
$H_1=-7$

$7,H_1,H_2,H_3,......H_{10},-\frac{1}{3}$ are in H.P

$\Rightarrow \frac{1}{7},\frac{1}{H_1},\frac{1}{H_2},\frac{1}{H_3},.........\frac{1}{H_{10}},-3$ are in A.P
If its common difference be $d$ then $-3$ is the 12th term $=\frac{1}{7}+11d$

$\Rightarrow 11d=-3-\frac{1}{7}=\frac{-22}{7}\Rightarrow d=\frac{-2}{7}$

$\therefore \frac{1}{H_1}=\frac{1}{7}-\frac{2}{7}=\frac{-1}{7}\Rightarrow H_1=-7$

$\frac{1}{H_2}=\frac{1}{7}-\frac{4}{7}=\frac{-3}{7}\Rightarrow H_2=\frac{-7}{3}$

$\frac{1}{H_{10}}=\frac{1}{7}-\frac{20}{7}=\frac{-19}{7}\Rightarrow H_{10}=\frac{-7}{19}$