There are two methods.
First method. Differences:
list them with an x where the ? is
9
19
40
x
146
Beside each of the the first four numbers write the difference between
the next number below it. That is,
Beside the 9 write 19-9 = 10
Beside the 19 write 40-19 = 21
Beside the 40 write x-40
Beside the x write 146-x which is 146-x
The numbers in the second column below are called "the first differences".
9 10
19 21
40 x-40
x 146-x
146
Do the same as before:
Beside the 10 write 21-10 = 11
Beside the 21 write (x-40)-21 = x-40-21 = x-61
Beside the x-40 write (146-x)-(x-40) = 146-x-x+40 = 186-2x
The numbers in the third column below are called "the second differences".
9 10 11
19 21 x-61
40 x-40 186-2x
x 146-x
146
Beside the 11 write (x-61)-11 = x-61-11 = x-72
Beside the x-61 write (186-2x)-(x-61) = 186-2x-x+61 = 247-3x
The numbers in the fourth column below are called "the third differences".
9 10 11 x-72
19 21 x-61 247-3x
40 x-40 186-2x
x 146-x
146
Beside the x-72 write (247-3x)-(x-72) = 247-3x-x+72 = 319-4x
There is only one fourth differences. We set it equal to 0.
9 10 11 x-72 319-4x
19 21 x-61 247-3x
40 x-40 186-2x
x 146-x
146
Set 319-4x = 0
-4x = -319
x = 319/4
That's the missing value.
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Second method, find the polynomial equation for
9 = a1
19 = a2
40 = a3
x = a4
146 = a5
Assume
Substitute and
Substitute and
Substitute and
Substitute and
So we have the system of equations:
Solve that and get , , ,
So
becomes:
So the missing term a4 is found by substituting n=4
Edwin
