The difference between the reciprocals of two consecutive positive odd integers 2/15 . find the intergers

3 and 5

Step-by-step explanation:

consecutive odd numbers have a difference of 2 between them.

let the consecutive odd numbers be n and n + 2, then

the reciprocals are and

The difference between the reciprocals is then

- =

combine the fractions into a single fraction

=

= ( cross- multiply )

2n(n + 2) = 30 ( divide both sides by 2 )

n(n + 2) = 15

n² + 2n - 15 = 0 ← in standard form

(n + 5)(n - 3) = 0 ← in factored form

equate each factor to zero and solve for n

n + 5 = 0 ⇒ n = - 5

n - 3 = 0 ⇒ n = 3

but n > 0 ⇒ n = 3

The 2 integers are n = 3 and n + 2 = 3 + 2 = 5

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!