SOLUTION: a quadratic pattern has a second term equal to 1, a third term equal to -6 and fifth term equal to -14 1.1.1 calculate the second difference of this quadratic pattern 1.1.2

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Question 1177109: a quadratic pattern has a second term equal to 1, a third term equal to -6 and fifth term equal to -14
1.1.1 calculate the second difference of this quadratic pattern
1.1.2 hence or otherwise, calculate the first term of the pattern

Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


A quadratic sequence has a constant second difference. So this is what we know about the sequence:
  ___    1    -6    ___   -14    terms 1 through 5
     ___   -7    ___   ___       first differences
         x     x     x           constant second differences

We can get expressions for the other first differences using the one known first difference and the constant second differences:
  ___     1    -6     ___     -14    terms 1 through 5
     -7-x   -7    -7+x   -7+2x       first differences
          x     x      x             constant second differences

Now we can get two different expressions in x for the 4th term of the sequence, using the 3rd and 5th terms and the second differences. Find the value of x that makes both expressions the same.

4th term, using the 3rd term -6 and the common difference (-7+x): -13+x
4th term, using the 5th term -14 and the common difference (-7+2x): -7-2x

-13%2Bx+=+-7-2x
3x+=+6
x+=+2

Now we can use that value of x to fill in the array, working from the bottom up:
   8    1   -6   -11   -14    terms 1 through 5
     -9   -7   -5    -3       first differences
        2    2     2          constant second differences

ANSWERS: The constant second difference is 2; the first term is 8.