Question

How to find the sum of the harmonic sequence below?
$\dfrac{1}{3} + \dfrac{1}{6} + \dfrac{1}{9} + \dfrac{1}{{12}}$

Answer 1

Hint: According to given in the question we have to determine the sum of the harmonic sequence which is $\dfrac{1}{3} + \dfrac{1}{6} + \dfrac{1}{9} + \dfrac{1}{{12}}$. So, first of all we have to put all fractions over a common denominator. That denominator must be a multiple of the given ones since the fractions $\dfrac{1}{3},\dfrac{1}{6},\dfrac{1}{9},\dfrac{1}{{12}}$ are all in the lowest terms. The smallest whole number that is a multiple of all four denominators, called least common multiple. So we can choose that common denominator.
Now, we have to add the numerator and see if the resulting fraction can be reduced to lowest terms.

Complete step-by-step solution:
Step 1: First of all we have to put all fractions over a common denominator. That denominator must be a multiple of the given ones since the fractions $\dfrac{1}{3},\dfrac{1}{6},\dfrac{1}{9},\dfrac{1}{{12}}$ are all in the lowest terms.
So, we can see that $36$ is the L.C.M of denominator of the given fraction $\dfrac{1}{3},\dfrac{1}{6},\dfrac{1}{9},\dfrac{1}{{12}}$. So, $36$ is called the least common multiple, so we can choose that common denominator.
Step 2: $\dfrac{1}{3} \times \dfrac{{12}}{{12}} = \dfrac{{12}}{{36}}$
$ \Rightarrow \dfrac{1}{6} \times \dfrac{6}{6} = \dfrac{6}{{36}}$
$ \Rightarrow \dfrac{1}{9} \times \dfrac{4}{4} = \dfrac{4}{{36}}$
$ \Rightarrow \dfrac{1}{{12}} \times \dfrac{3}{3} = \dfrac{3}{{36}}$
Step 3: Now, we have to add the numerator and see if the resulting fraction can be reduced to lowest terms
$
   \Rightarrow \dfrac{{12}}{{36}} + \dfrac{6}{{36}} + \dfrac{4}{{36}} + \dfrac{3}{{36}} \\
   \Rightarrow \dfrac{{12 + 6 + 4 + 3}}{{36}} \\
   \Rightarrow \dfrac{{25}}{{36}}
 $
 The sum cannot be reduced to lower terms because 25 and 36 have no common factors greater than one.

Hence, the sum of the harmonic sequence as $\dfrac{1}{3} + \dfrac{1}{6} + \dfrac{1}{9} + \dfrac{1}{{12}}$ is $\dfrac{{25}}{{36}}$.

Note: It is necessary to put all fractions over a common denominator. That denominator must be a multiple of the given ones since the fractions $\dfrac{1}{3},\dfrac{1}{6},\dfrac{1}{9},\dfrac{1}{{12}}$ are all in the lowest terms.
It is necessary to take the L.C.M of denominator of the given fraction. So that L.CM. Is called the least common multiple, so we can choose that common denominator.