Unknown

7.1. Interpolation and Extrapolation 207

Then verify that p(t) = CzO bipi(t) is the desired polynomial. This

is called the Lagrange polynomial determined by the conditions.

6. Determine the quadratic Lagrange polynomial Q which satisfies
   q(-2) = 7, n(l) = 2 and q(3) = 1. Write it in the form at2 + bt + c.


7. In practice, the Lagrange form is not generally the best way to obtain
   a polynomial fitting certain data, particularly if many other values of
   the polynomial are also required.
   Consider the sequence: 1, 3, 6, 10, 15, 21, 28, 36,.... Show that
   these are successive values of a quadratic polynomial evaluated on
   the positive integers.
   We form a di$erence table for this sequence as follows. Write out
   the terms of the sequence in a column; beside the column, put a
   second whose entries are the differences between successive terms of
   the first; form a third column from the second in the same fashion,
   and continue on. Here is what you get in this case:

1

2

3 1

3 0

6 1

4 0

10 1

5 0

15 1

From the table, decide what terms should follow the entry 36 in the

original sequence. Check your guess using the quadratic polynomial.

8. Consider the sequence of values: 2, 11,35,85, 175,322,546,870,....
   Carry out the procedure of Exercise 7, continuing until a column of
   zeros is obtained. This will occur in the sixth column of the table.
   Let us suppose that the sequence arises from the evaluation of a
   function f(n) at the successive points 1, 2, 3,.... What do you think
   the form of this function might be? What do you think the next two
   terms of the sequence following 870 might be?
   What is the nth term of the sixth column of the table?
   What is the nth term of the fifth column of the table?
   What is the nth term of the fourth column of the table? the third
   column? the second column? the first column?
   Give a function f(n) which reproduces the sequence.