Question

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

Answer 1

Let one of the odd numbers be x

Since they're consecutive therefore the other number would be x+2

Given sum of the reciprocals = 12/35

=> 1/x + 1/x+2 = 12/35

$\frac{1}{x} + \frac{1}{x + 2} = \frac{12}{35} \\ \frac{1}{x} + \frac{1}{x + 2} - \frac{12}{35} = 0 \\ \frac{35(x + 2) + 35x - 12x(x + 2)}{35x(x + 2)} = 0 \\ \frac{35x + 70 + 35x - 12 {x}^{2} - 24x }{35(x + 2)} = 0 \\ \frac{46x + 70 - 12 {x}^{2} }{35(x + 2)} = 0 \\ 46x + 70 - 12 {x}^{2} = 0 \\ - 12x {}^{2} + 46x + 70 = 0 \\ dividing \: both \: sides \: by \: - 2 \\ 6x {}^{2} - 23x - 35 = 0 \\ x = \frac{ - ( - 23) + \sqrt{( - 23) {}^{2} - 4 \times 6 \times ( - 35) } }{2 \times 6} \\ (by \: using \: quadratic \: formula \: also \: only \: taking \: + ve \: \: value) \\ x = \frac{23 + \sqrt{1369} }{12} \\ x = \frac{23 + 37}{12} \\ x = \frac{60}{12} \\ x = 5$

x = 5

therefore, one of the odd numbers is 5 and the other is 5+2 = 7