Question

Find two H.M's between 1/2 , 4/17 .
Ans: 4/11, 2/7

Answer 1

Hint

If the multiplicative inverses of each sequence form an A.P, the sequence is called a harmonic sequence.

Solution

Let $\{a_{n}\}$ be the harmonic sequence. Then, $\left\{\dfrac{1}{a_{n}} \right\}$ is an arithmetic sequence.

The two arithmetic means between $2,\dfrac{17}{4}$ can be found by forming an arithmetic sequence.

$\implies \dfrac{1}{a_{n}}=2+(n-1)d$

For n=1 and n=4, we have $\dfrac{1}{a_{1}}=2$ and $\dfrac{1}{a_{4}}=2+3d$.

To form an arithmetic sequence, let's establish a system equation.

Equation, $\dfrac{1}{a_{4}} =\dfrac{17}{4}$

$\implies 2+3d=\dfrac{17}{4}$

$\implies 3d=\dfrac{9}{4}$

$\implies d=\dfrac{3}{4}$

The arithmetic sequence is $\dfrac{1}{a_{n}} =2+\dfrac{3}{4} (n-1)$. Hence $\dfrac{1}{a_{2}} =\dfrac{11}{4}$ and $\dfrac{1}{a_{3}} =\dfrac{7}{2}$.

Now we are required to find the harmonic sequence $\{a_{n}\}$. The required answer is $\boxed{a_{2}=\dfrac{4}{11}}$ and $\boxed{a_{3}=\dfrac{2}{7}}$.

So, $\boxed{\dfrac{1}{2} ,\dfrac{4}{11} ,\dfrac{2}{7} ,\dfrac{4}{17}}$ are in a harmonic sequence.

Extra information

Refer to the attachment for more information. Here,

R.M.S, root mean square, $\sqrt{\dfrac{a^2+b^2}{2} }$.

A.M, arithmetic mean, $\dfrac{a+b}{2}$.

G.M, geometric mean, $\sqrt{ab}$.

H.M, harmonic mean, $\dfrac{2ab}{a+b}$.

The lengths of each line segment represent R.M.S, A.M, G.M, H.M.

• Blue line segment: R.M.S
• Red line segment: A.M
• Green line segment: G.M
• Purple line segment: H.M

The inequality R.M.S ≥ A.M ≥ G.M ≥ H.M is satisfied for two positive numbers $a$ and $b$. A.M ≥ G.M is frequently used in geometry or algebra.