List them in a vertical column:

 10 
 18 
 40 
 78 
112

Form a second column in this fashion:
To the right of each number in the 1st column 
except the bottom one, subtract it from the number
just beneath it:

 10   8  
 18  22 
 40  38
 78  34
112
 
Form a third column in this fashion:
To the right of each number in the 2nd column 
except the bottom one, subtract it from the number
just beneath it:

 10   8  14
 18  22  16
 40  38  -4
 78  34
112

Form a fourth column in this fashion:
To the right of each number in the 3rd column 
except the bottom one, subtract it from the number
just beneath it:

 10   8  14   2
 18  22  16 -20 
 40  38  -4
 78  34
112

The fifth column will consist of only one number,
-20 - 2 = -22

 10   8  14   2  -22
 18  22  16 -20 
 40  38  -4
 78  34
112

Next duplicate the -22 just below the -22

 10   8  14   2  -22
 18  22  16 -20  -22 
 40  38  -4
 78  34
112

Now work backwards to place a new number at the bottom
of each column so that the difference between it and the
number just above it will be the number just to the right
of the one above it.  That will amount to adding it to
the number just left of it and placing the result below
that number.  So we begin by adding -22 to -20,
getting -42 and writing this at the bottom of the 4th column

 10   8  14   2  -22
 18  22  16 -20  -22 
 40  38  -4 -42
 78  34
112

Next we add -42 to -4, getting -46 and we write this at the 
bottom of the 3rd column:

 10   8  14   2  -22
 18  22  16 -20  -22 
 40  38  -4 -42
 78  34 -46
112

Next we add -46 to 34, getting -12 and we write this at the 
bottom of the 2nd column:

 10   8  14   2  -22
 18  22  16 -20  -22 
 40  38  -4 -42
 78  34 -46
112 -12

Finally we add -12 to 112, getting 100 and we write this at the 
bottom of the 1st column:

 10   8  14   2  -22
 18  22  16 -20  -22 
 40  38  -4 -42
 78  34 -46
112 -12
100

So the next term is 100.  The sequence goes:

10, 18, 40, 78, 112, 100

Edwin