Hiii Friends...
Q
show that the product of two consecutive positive integers is divisible by 2
Answers
Answer:
Let the two consecutive positive integers be x and x+1
Let x=2a (even) and x+1 =2a+1(odd)
According to the given question
Product of two integers (x)(x+1)
=x^{2}+x=x
2
+x
Case 1: if x is an even number
By putting the values
We have x^{2}+x=2 a^{2}+2 ax
2
+x=2a
2
+2a
=2a (a+1)
Check and divide the above expression by 2
= 2a (a+1)/2
We get a (a+1)
Hence it is clearly proved that the product of 2 integers are divisible by 2
Case 2: If x is an odd number
By putting the values x=2a+1
We have x^{2}+x=(2 a+1)^{2}+(2 a+1)x
2
+x=(2a+1)
2
+(2a+1)
\begin{lgathered}\begin{array}{l}{=4 a^{2}+1+4 a+2 a+1} \\ {=4 a^{2}+6 a+2} \\ {=2\left(2 a^{2}+3 a+1\right)}\end{array}\end{lgathered}
=4a
2
+1+4a+2a+1
=4a
2
+6a+2
=2(2a
2
+3a+1)
Check: Divide the above expression by 2
=2\left(2 a^{2}+3 a+1\right) / 2=2(2a
2
+3a+1)/2
We get 2 a^{2}+3 a+12a
2
+3a+1
Hence proved