N is a positive integer i) Explain why n(n+1) must be an even number?
ii) Explain why 2n-1 must be an odd number?

Step-by-step explanation:

product of two consecutive numbers is always even.

n(n+1)

case 1

if n is odd,let n=2k+1,where k is an integer

n+1=2k+1+1=2k+2=2(k+1)

n(n+1)=(2k+1)[2(k+1)]=2(2k+1)(k+1)=even number.

case 2

if n is even,let n=2m,where m is an integer.

n+1=2m+1

n(n+1)=2m(2m+1)=even integer.

so in both cases n(n+1) is even.

b = 10

Step-by-step explanation:

The expression 9x² - 100 is a difference of squares. It factors into the form (3x -b)(3x+b). Here b is the square root of 100. The square root is 10.

The factors are (3x -10)(3x+10).