The next number is 4. The one after that is -27.

These numbers are the result of evaluating the function

f[x] = (-6 + 2x + 7x^2 -x^3)/2

for x = 1 through 7

_____

Problems of this sort can often be worked by writing differences. The "first differences" are computed by subtracting each number from the next. "Second differences" are computed the same way from the first differences. Likewise, third-differences. You are looking for a row that is constant. When you find that, you can work back up the chain to determine the next number in the sequence.

1st: 8 9 7 2 -6 (-17)

2nd: 1 -2 -5 -8 (-11)

3rd: -3 -3 -3 (-3)

Here, we have computed that the next number is 21 - 17 = 4.

Once you have this, you can write the equation from the first number of each of the sequence and the rows of differences. (From the top down, these are 1, 8, 1, -3.)

The equation is the summation of terms ...

f[x] = (sequence starter)

+ (first difference)(x-1)/1!

+ (second difference)(x-1)(x-2)/2!

+ (third difference)(x-1)(x-2)(x-3)/3!

+ {similar terms for succeeding differences}

You will note that when the difference of differences is zero, the term of f[x] goes to zero. Thus, the last row of differences of any consequence is the one where the differences are constant.

This function assumes that x=1 for the first number of the series. You can use what you learned about transforming functions if you wish to have the sequence start with x=0 instead.

These numbers are the result of evaluating the function

f[x] = (-6 + 2x + 7x^2 -x^3)/2

for x = 1 through 7

_____

Problems of this sort can often be worked by writing differences. The "first differences" are computed by subtracting each number from the next. "Second differences" are computed the same way from the first differences. Likewise, third-differences. You are looking for a row that is constant. When you find that, you can work back up the chain to determine the next number in the sequence.

1st: 8 9 7 2 -6 (-17)

2nd: 1 -2 -5 -8 (-11)

3rd: -3 -3 -3 (-3)

Here, we have computed that the next number is 21 - 17 = 4.

Once you have this, you can write the equation from the first number of each of the sequence and the rows of differences. (From the top down, these are 1, 8, 1, -3.)

The equation is the summation of terms ...

f[x] = (sequence starter)

+ (first difference)(x-1)/1!

+ (second difference)(x-1)(x-2)/2!

+ (third difference)(x-1)(x-2)(x-3)/3!

+ {similar terms for succeeding differences}

You will note that when the difference of differences is zero, the term of f[x] goes to zero. Thus, the last row of differences of any consequence is the one where the differences are constant.

This function assumes that x=1 for the first number of the series. You can use what you learned about transforming functions if you wish to have the sequence start with x=0 instead.