Question 208402
<font size = 9 color="red">Edwin's solution:</b></pre></font>
Find an equation for the points (1,1) (2,3) (3,6) and (4, 10)
<pre><font size =4 color = "indigo"><b>
Put down the y values in a row:

   1   3   6   10

Find the first successive differences by subtracting 
neighboring numbers, and write their differences 
between and below them:

   1   3   6   10
     2   3   4

Find the second successive differences again by 
subtracting neighboring numbers, and writing their 
differences between and below them:

   1   3   6   10
     2   3   4
       1   1

That took 2 lines of differences until all the differences
were all the same, in this case, 1. 

So we know that we need an equation of degree 2.

{{{Ax^2+Bx+C=y}}}

Plug in all three points:

Plug in (1,1) 

{{{A(1)^2+B(1)+C=1}}}
{{{A+B+C=1}}}

Plug in (2,3) 

{{{A(2)^2+B(2)+C=3}}}
{{{4A+2B+C=3}}}

Plug in (3,6) 

{{{A(3)^2+B(3)+C=6}}}
{{{9A+3B+C=6}}}

You don't need the fourth point.

So you have the system:

{{{system(A+B+C=1,4A+2B+C=3,9A+3B+C=6)}}}

Solve that system and get:

{{{A=1/2}}}, {{{B=1/2}}} and {{{C=0}}}

Therefore:

{{{Ax^2+Bx+C=y}}}

becomes:

{{{(1/2)x^2+(1/2)x+(0)=y}}}

or

{{{y=(1/2)x^2+(1/2)x}}}

Factor out {{{(1/2)x}}} on the right:

{{{y=(1/2)x(x+1)}}}

To take {{{1/2}}} is the same as dividing by {{{2}}}:

{{{y=(x(x+1))/2}}}

Edwin</pre>