Question

The sum of the reciprocals of two consecutive odd natural numbers is12/35. Find the numbers

Answer 1

Answer:

Required consecutive odd numbers are 7 and 5.

Step-by-step explanation:

Let the required consecutive odd natural numbers are 2a + 1 and 2a + 3 .

According to the question : -

⇒ Sum of the reciprocal of the numbers is 12 / 35.

$\implies \dfrac{1}{2a+1}+\dfrac{1}{2a+3}=\dfrac{12}{35}$

$\implies\dfrac{(2a+3)+(2a+1)}{(2a+1)(2a+3)}=\dfrac{12}{35}$

$\implies\dfrac{2a+3+2a+1}{4a^2+6a+2a+3}=\dfrac{12}{35}$

$\implies \dfrac{4a+4}{4a^2+8a+3}=\dfrac{12}{35}$

$\implies\dfrac{4(a+1)}{4a^2+8a+3}=\dfrac{12}{35}$

$\implies \dfrac{a+1}{4a^2+8a+3}=\dfrac{3}{35}$

$\implies 35(a+1)=3(4a^2+8a+3)$

$\implies 35a + 35 = 12a^2 + 24a + 9$

$\implies 12a^2 + 24a - 35a + 9 - 35=0$

$\implies 12a^2 - 11a - 26 = 0$

$\implies 12a^2 - ( 24 - 13 )a - 26 = 0$

$\implies 12a^2 - 24a + 13a - 26 = 0$

$\impies 12a( a - 2 ) + 13( a - 2 ) = 0$

$\implies (a-2)(12a+13)=0$

Therefore, by Zero Product Rule : -

a - 2 = 0 ⇒ a = 2

12a + 13 = 0 ⇒ a = -  13 / 12

It is given that the numbers are natural numbers, so they cannot be negative.

So,

a = 2

Required numbers are : -

2a + 1 = 2( 2 ) + 1 = 4 + 1 = 5

2a + 3 = 5 + 2 = 7