prove that the product of two consecutive natural numbers is an even umber using the principal of mathematical induction
let n and (n+1) be two consecutive natural numbers.
here we are require to prove that n.(n+1) is an even number.
to prove this by mathematical induction method, we will follow three steps.
first we will show that it is true for n =1 and then we will assume that it is true for n = k
with the help of assumption we will try to prove that the given condition also holds for n = (k+1)
step 1: for n =1
n.(n+1)= 1.2 = 2 is an even number
step 2: let us assume that k(k+1) is an even number. i.e. either k or (k+1) is an even number.
step 3: for (k+1)(k+2) , by the assumption two cases arise:
case I: if k is an even number, i.e. k =2m , where m is an integer
then k+2=2m+2=2(m+1) is also an even number, therefore (k+1).(k+2) is also an even number
case II: if (k+1) is an even number. (k+1).(k+2) is an even number, as the multiplication of an even number with any other integer result into an even integer.
thus (k+1)(k+2) is an even integer. thus it is true for any integer n.
thus the product of two consecutive natural number is an even integer.
hope this helps you
here we are require to prove that n.(n+1) is an even number.
to prove this by mathematical induction method, we will follow three steps.
first we will show that it is true for n =1 and then we will assume that it is true for n = k
with the help of assumption we will try to prove that the given condition also holds for n = (k+1)
step 1: for n =1
n.(n+1)= 1.2 = 2 is an even number
step 2: let us assume that k(k+1) is an even number. i.e. either k or (k+1) is an even number.
step 3: for (k+1)(k+2) , by the assumption two cases arise:
case I: if k is an even number, i.e. k =2m , where m is an integer
then k+2=2m+2=2(m+1) is also an even number, therefore (k+1).(k+2) is also an even number
case II: if (k+1) is an even number. (k+1).(k+2) is an even number, as the multiplication of an even number with any other integer result into an even integer.
thus (k+1)(k+2) is an even integer. thus it is true for any integer n.
thus the product of two consecutive natural number is an even integer.
hope this helps you