Question

The sum of the reciprocals of two consecutive odd integers is −16
63
. Find the two numbers.

Answer 1

Step-by-step explanation:

nx

2

+x(1+3+5+...+(2n−1))+(1.2+2.3+...+(n−1).n)−10n=0

⇒nx

2

+xn

2

+

3

n(n

2

−1)

−10n=0

⇒x

2

+nx+

3

(n

2

−1)

−10=0

Let α and β be the roots of the equation then α+β=−n and αβ=

3

(n

2

−1)

−10

And (α−β)

2

=1

⇒(α+β)

2

−4αβ=1

⇒n

2

−4(

3

(n

2

−1)

−10)=1

⇒3n

2

−4n

2

+4+120−3=0

⇒n

2

=121

∴n=11