“table de différence” Réponses codées

table de différence

We make what is called a "difference table":

Write the numbers in a row:
              
            9   73    241    561    1081    1849 

Under that row, subtract each adjacent pair of numbers,
placing the difference between and below them.

(73-9=64, so write 64 between and below the pair 9  73,
 241-73=168, so write 168 between and below the pair 73   241, etc.) 

Like this:

            9   73     241   561    1081    1849 
              64    168   320    520     768 
                 
That row is called the row of 1st differences.                    
Under that row, again subtract each adjacent pair of numbers,
placing the difference between and below them, like this:


            9   73     241   561    1081    1849 
              64    168   320    520     768 
                 104    152   200     248

That row is called the row of 2nd differences.                  
Under that row, again subtract each adjacent pair of numbers,
placing the difference between and below them, like this:

            9   73     241   561    1081    1849 
              64    168   320    520     768 
                 104    152   200     248
                     48     48     48

That row is called the row of 3rd differences.
Now notice that they are all the same, so we don't need to
make any more rows of differences.  So we assume that the next 
3rd difference, if we had the next number after 1849, would also 
be a 48, so we write another 48 on the bottom row.
(I'll color it red):


            9   73     241   561    1081    1849 
              64    168   320    520     768 
                 104    152   200     248
                     48     48     48     48

Then we work backwards by adding.  Add the 48 to the 248, getting 296,
Write that to the right of the 248. (I'll color it red too).

            9   73     241   561    1081    1849 
              64    168   320    520     768 
                 104    152   200     248    296
                     48     48     48     48 

Still working backward, Add the 296 to the 768, getting 1064,
Write that to the right of the 768. (I'll color it red too).

            9   73     241   561    1081    1849 
              64    168   320    520     768     1064
                 104    152   200     248    296
                     48     48     48     48

One more step working backwards.  Add 1064 to the 1849, getting
2913, write it at the end of the first row, and there is your answer!:

            9   73     241   561    1081    1849     2913  
              64    168   320    520     768     1064
                 104    152   200     248    296
                     48     48     48     48

Answer: The next term is 2913

Edwin