If a and b are two odd positive integers such that a>b,then prove that

1.	If a and b are two odd positive integers such that a>b,then prove that one of the
Answer» If a and b are odd numbers then it should be in 2q+1 or 2q+3 form where\xa0q is a positive integer.Let a = 2q + 3 , b = 2q + 1 and\xa0a > bNow,\xa0{tex}\\frac{a + b } {2} = \\frac{ 2q + 3 + 2q + 1}{2}{/tex}{tex}= \\frac { 4 q + 4 } { 2 }{/tex}= 2q + 2{tex}\\frac{a+b}{2}{/tex}=2(q+1)\xa0= an even number..........(1)Now\xa0{tex}\\frac { a - b } { 2 } = \\frac { ( 2 q + 3 ) - ( 2 q + 1 ) } { 2 }{/tex}{tex}= \\frac { 2 q + 3 - 2 q - 1 } { 2 }{/tex}{tex}\\frac{a-b}{2}= \\frac { 2 } { 2 }{/tex}\xa0= 1 = an odd number..........(2)Hence From (1) and (2)\xa0{tex}\\frac{a+b}{2}{/tex} and\xa0{tex}\\frac{a-b}{2}{/tex} are even and odd numbers respectively

Discussion