Question

X=
`(2n+6)/(n-2)` x ∈ N, n ∈ N

Answer 1

x = {3,4,7,12}

n = {12,7,4,3}

(upside down)

Explanation:

N = {1,2,3,4,5....}

`\:`

if n = 1

`\sf{x = (2n+6)/(n-2)}`

`\sf{x = (2(1) + 6)/(1 - 2)}`

`\sf{x = (2 + 6)/(-1)}`

`\sf{x = (8)/(-1)}`

`\sf{x = - 8}`

• x not N

`\:`

if n = 2

`\sf{x = (2n+6)/(n-2)}`

`\sf{x = (2(2) +6)/(2-2)}`

`\sf{x = (4 + 6)/(2 - 2)}`

`\sf{x = (8)/(0)}`

`\sf{x = \infty }`

• x not N

`\:`

if n = 3

`\sf{x = (2n+6)/(n-2)}`

`\sf{x = (2(3) +6)/(3-2)}`

`\sf{x = (6 + 6)/(1)}`

`\sf{x = (12)/(1)}`

`\sf{x = 12}`

• x is N

`\:`

if n = 4

`\sf{x = (2n+6)/(n-2)}`

`\sf{x = (2(4) +6)/(4-2)}`

`\sf{x = (8 + 6)/(2)}`

`\sf{x = (14)/(2)}`

`\sf{x = 7}`

• x is N

`\:`

if n = 5

`\sf{x = (2n+6)/(n-2)}`

`\sf{x = (2(5) +6)/(5-2)}`

`\sf{x = (10 + 6)/(3)}`

`\sf{x = (16)/(3)}`

• x not N

`\:`

if n = 6

`\sf{x = (2n+6)/(n-2)}`

`\sf{x = (2(6) +6)/(6-2)}`

`\sf{x = (12 + 6)/(4)}`

`\sf{x = (18)/(4)}`

• x not N

`\:`

if n = 7

`\sf{x = (2n+6)/(n-2)}`

`\sf{x = (2(7) +6)/(7-2)}`

`\sf{x = (14 + 6)/(5)}`

`\sf{x = (20)/(5)}`

`\sf{x = 4}`

• x is N

`\:`

if n = 12

`\sf{x = (2n+6)/(n-2)}`

`\sf{x = (2(12) +6)/(12-2)}`

`\sf{x = (24 + 6)/(10)}`

`\sf{x = (30)/(10)}`

`\sf{x = 3}`

• x is N

`\:`

conclusion

• I found the pattern of numbers that the value of n is 3, 4, 7,12
• For x, I found the pattern of numbers that the value of x is 12, 7, 4,3