4.4 Q-4 | Sequence And Series | Mathematics 11

Exercise 4.4 — Question 4

A harmonic sequence is a sequence whose reciprocals form an arithmetic sequence.

If $a_{1}, a_{2}, a_{3}, \dots$ is a harmonic sequence, then $\frac{1}{a _{1}}, \frac{1}{a _{2}}, \frac{1}{a _{3}}, \dots$ is an arithmetic sequence.

General term of a Harmonic Sequence:

If the corresponding arithmetic sequence has first term $a$ and common difference $d$ , then the $n$ th term of the harmonic sequence is:

$T_{n} = \frac{1}{a + ( n - 1 ) d}$

Harmonic Mean (H.M.) between two numbers $a$ and $b$ :

$H = \frac{2 ab}{a + b}$

Problem: Find the $n$ th term of the harmonic sequence: $\frac{1}{3}, \frac{1}{7}, \frac{1}{11}, \dots$

Solution:

Step 1: Take reciprocals to get the corresponding arithmetic sequence: $3, 7, 11, \dots$

Step 2: Identify $a = 3$ and $d = 7 - 3 = 4$ .

Step 3: The $n$ th term of the arithmetic sequence is: $a_{n} = a + (n - 1) d = 3 + (n - 1) (4) = 4 n - 1$

Step 4: Therefore, the $n$ th term of the harmonic sequence is: $T_{n} = \frac{1}{4 n - 1}$

For two positive numbers $a$ and $b$ :

Important inequality: $A \geq G \geq H$

Also: $G^{2} = A \cdot H$