MY INNOVATION

GOODBYE TO LABORIOUS METHOD

My project is a simple way to find the difference between the cubes of two consecutive integers.

         If we want to find the difference between two cubes we should do a laborious method. For example,

                 23³-22³= (23×23×23)-(22×22×22)

Instead of doing all these laborious method, to calculate the difference between the cubes of two consecutive integers we can do a simple method.

That is,

    

For any consecutive integers a&b

If a>b then,

a³-b³=3ab+1

Examples:

·    3³-2³=27-8=19

   3ab+1=3×6+1=19

·    205³-204³=8615125-8489664=125461

 3ab+1=3×41820+1=125461

·    -100³—101³=-1000000-1030301=30301

 3ab+1=3×10100+1=30301

For any consecutive integers a&b If b>a then, a³-b³=-(3ab+1)

 

 Examples:

·      2³-3³=8-27= -19

 - (3ab+1)=-(3×6+1)= -19

·     -101³—100³=-1030301+100000= -30301

- (3ab+1)=-(3×10100+1)= -30301

I have also findout a simple method to check whether the sum of two cubes is right. That is,

a²+b²=2ab+ (a-b) ²

Examples:

·     4²+2²=16+4=20

  2ab+(a-b)²=2×4×2+ (4-2)²=16+4=20