Question

Arithmetic mean between two numbers is 5 and Geometric mean between them is 4. Find the Harmonic mean between the numbers.
And in a harmonic progression third term and fifth terms are respectively 1 and $\frac{1}{-5}$. Find the tenth term.

Answer 1

First Part,

Let $a$ and $b$ be two numbers.

According to question,

$\frac{a+b}{2}=5⟹a+b=\mathrm{10.....................}\left(i\right)$

$\sqrt{ab}=4⟹ab=\mathrm{16.........................}\left(ii\right)$

Now, H.M between $a$ and $b=\frac{2ab}{a+b}$

$=\frac{2\times 16}{10}$

$=\frac{16}{5}$

Or,

Let $a$ be first term and $d$ be the common difference of corresponding A.P.

$\frac{1}{a+2d}=\mathrm{1.............}\left(i\right)$

$\frac{1}{a+4d}=-\mathrm{5....................}\left(ii\right)$

Solving both equations , we get

$a=7$ and $d=-3$

Now, $t_{10}=\frac{1}{a+9d}=\frac{1}{7+9\times (-3)}=-\frac{1}{20}$