Math, asked by haideracttpdjqyq, 1 year ago

The product of any three consecutive numbers is always divisible by 6.give examples and justify your answer

Answers

Answered by sonabrainly
10

Here is a formal proof with all the steps very explicitly outlined. The proper statement of this theorem is as follows:


For every natural number k greater than or equal to 1, there exists such a natural number a that:


k(k+1)(k+2)=6a


This can be proved by induction, for k=1:


1(1+1)(1+2)=6a

6=6a

a=1


For k=n+1 the theorem states:


(n+1)(n+2)(n+3)=6a


To use the induction hypothesis, expand the multiplication using the last bracket:


(n+1)(n+2)(n+3)=

(n+1)(n+2)n+(n+1)(n+2)3=

n(n+1)(n+2)+3(n+1)(n+2)


By the induction hypothesis, n(n+1)(n+2) is equal to 6b for some natural number b. (n+1)(n+2) is a product of an odd number and even number, hence a number of the form 2c for some other natural number c:


n(n+1)(n+2)=6b

(n+1)(n+2)=2c


Now just substitute this into the previous expansion:


(n+1)(n+2)(n+3)=

n(n+1)(n+2)+3(n+1)(n+2)=

6b+3*2c=

6b+6c=

6(b+c)

a=b+c


So, finally:


(n+1)(n+2)(n+3) = 6[(n(n+1)(n+2))/6+((n+1)(n+2))/2]


With (n(n+1)(n+2))/6+((n+1)(n+2))/2 being a natural number, which was to be proven. This proof also illustrates the fact that, having a triple of consecutive numbers and their product, adding to this product the product of the last two numbers and the number 3 results in the product of the next three consecutive numbers, e.g.:


1*2*3=6

2*3*4=24=6+3*(2*3)

3*4*5=60=24+3*(3*4)

4*5*6=120=60+3*(4*5)

5*6*7=210=120+3*(5*6)

...

Answered by gatesindia1
14

Answer:

Hey guys your answer is below!!

Step-by-step explanation:

As you all know consecutive numbers are numbers which follow each other in order, without gaps, from smallest to largest .

For example 2,3 and 4

2 x 3 x 4 x = 24

So 24 is divisible by 6 ( 6 x 4)

Similar questions