How to Get Value of 11^5 after 11^4 from Pascal Triangle.

[mathspassion]

Edited January 6, 2017 by mathspassion

[imatfaal]

Again - similar to your last "discovery" - this is a simple artifact of the number system and multiplication.

 

If you look at the basis of Pascal and Binomial expansion for (a+x)^6 to extend you get as follows

 

[latex]1a^6x^0+6a^5x^1+15a^4x^2+20a^3x^3+15a^2x^4+6a^1x^5+1a^0x^6[/latex]

 

If you set a to be 10 and x to be 1 (ie a is the tens and x is the units in decimal counting) this simplifies to

 

[latex]1\cdot10^6\cdot1+6\cdot10^5\cdot1+15\cdot10^4\cdot1+20\cdot10^3\cdot1+15\cdot10^2\cdot1+6\cdot10^1\cdot1+1\cdot10^0\cdot1[/latex]

 

simplify again

 

[latex]1\cdot10^6+6\cdot10^5+15\cdot10^4+20\cdot10^3+15\cdot10^2+6\cdot10^1+1[/latex]

 

and again

 

1,000,000 + 600,000 + 150,000 + 20,000 + 1,500 + 60 + 1 = 1,771,561 = 11^6

 

It works with any number and any power - it is just more obvious when both a and x are one